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On positive definite thresholding of correlation matrices

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.

On positive definite thresholding of correlation matrices

Abstract

Standard thresholding techniques for correlation matrices often destroy positive semidefiniteness. We investigate the construction of positive definite functions that vanish on specific sets , 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 , any soft-thresholding operator that preserves positive semidefiniteness necessarily induces a geometric collapse of the feature space, as quantified by an bound on the faithfulness constant. Such demonstrates that geometrically unbiased soft-thresholding limits the recoverable signal.
Paper Structure (15 sections, 9 theorems, 33 equations, 2 figures)

This paper contains 15 sections, 9 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Kernel embeddings
  4. Spherical Harmonics and Gegenbauer Polynomials
  5. Darboux estimates
  6. Derivative Estimates
  7. Three-term recurrence
  8. Delsarte's method and faithfulness
  9. Existence and structural bounds
  10. Explicit faithfulness bounds and the two-point gap
  11. One Point Thresholding
  12. Two Point Thresholding
  13. Interval thresholding
  14. Conjectures and remarks
  15. Clustering and Sparsity

Key Result

Theorem 2.1

Let $K$ be a positive definite kernel on a set $X$. Then there exists a essentially unique Hilbert space $\mathcal{H}$ and a feature map $\iota: X \to \mathcal{H}$ such that the kernel perfectly recovers the inner product in $\mathcal{H}$:

Figures (2)

  • Figure 1: The thresholding function for $K = \{0.1\}, n=3$
  • Figure 2: The thresholding function for $K = \{\pm 0.1\}, n=3$

Theorems & Definitions (26)

  • Definition 1.1
  • Remark 1.2
  • Theorem 2.1: Aronszajn's theorem paulsen2016introduction
  • Lemma 2.2
  • proof
  • Theorem 3.1: Delsarte's Estimate
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 16 more