On positive definite thresholding of correlation matrices
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911506486001664 |
|---|---|
| author | Damase, Sujit Sakharam Pascoe, James Eldred |
| author_facet | Damase, Sujit Sakharam Pascoe, James Eldred |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11040 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On positive definite thresholding of correlation matrices Damase, Sujit Sakharam Pascoe, James Eldred Statistics Theory Information Theory Classical Analysis and ODEs Functional Analysis Metric Geometry 62H12, 43A35, 46E22, 15B48, 33C45 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. |
| title | On positive definite thresholding of correlation matrices |
| topic | Statistics Theory Information Theory Classical Analysis and ODEs Functional Analysis Metric Geometry 62H12, 43A35, 46E22, 15B48, 33C45 |
| url | https://arxiv.org/abs/2603.11040 |