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The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

Apoorva Khare

Abstract

We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg, Rudin, and Loewner, the subject has attracted renewed attention owing to its connections to various applied fields and techniques. In this survey we will focus mainly on the question of preserving positivity in all dimensions. Connections to distance geometry and metric embeddings, positive definite sequences and functions, Fourier analysis, applications and covariance estimation, Schur polynomials, and finite fields will be discussed. The Appendix contains a mini-survey of sphere packings, kissing numbers, and their "lattice" versions. This part overlaps with the rest of the article via Schoenberg's classification of the positive definite functions on spheres, aka dimension-free entrywise positivity preservers with a rank constraint - applied via Delsarte's linear programming method.

The entrywise calculus and dimension-free positivity preservers, with an Appendix on sphere packings

Abstract

We present an overview of a classical theme in analysis and matrix positivity: the question of which functions preserve positive semidefiniteness when applied entrywise. In addition to drawing the attention of experts such as Schoenberg, Rudin, and Loewner, the subject has attracted renewed attention owing to its connections to various applied fields and techniques. In this survey we will focus mainly on the question of preserving positivity in all dimensions. Connections to distance geometry and metric embeddings, positive definite sequences and functions, Fourier analysis, applications and covariance estimation, Schur polynomials, and finite fields will be discussed. The Appendix contains a mini-survey of sphere packings, kissing numbers, and their "lattice" versions. This part overlaps with the rest of the article via Schoenberg's classification of the positive definite functions on spheres, aka dimension-free entrywise positivity preservers with a rank constraint - applied via Delsarte's linear programming method.

Paper Structure

This paper contains 41 sections, 61 theorems, 68 equations, 2 figures, 2 tables.

Table of Contents

  1. Introducing positivity preservers
  2. Schoenberg's theorem and its (classical) variants
  3. Matrices with entries in $[-1,1]$
  4. Positive definite sequences, Toeplitz matrices on the circle, and Rudin's result
  5. Open intervals; Hankel matrices
  6. Complex domains
  7. Schoenberg's motivations: distance geometry
  8. Metric embeddings
  9. Positive definite functions
  10. Reconciling the two notions of positive definiteness
  11. Schoenberg: from positive definite functions on spheres to positivity preservers
  12. From Schoenberg's positive definite functions to sphere packings
  13. Modern motivations; fixed dimension
  14. An application to machine learning
  15. Rudin vs. Schoenberg
  16. ...and 26 more sections

Key Result

Theorem 1.1

The following are equivalent for a complex (resp. real) Hermitian matrix $A_{n \times n}$:

Figures (2)

  • Figure 4.1: Math-Genealogy of some of the experts in positivity, its preservers, and connections
  • Figure A.1: The nodes (or simple roots) of the $E_8$ Dynkin diagram

Theorems & Definitions (113)

  • Theorem 1.1
  • Lemma 1.2
  • Definition 1.3
  • Proposition 1.4
  • Theorem 1.5: Hansen92
  • Theorem 2.1: Schur product theorem, Schur1911
  • proof
  • Definition 2.2
  • Theorem 2.3: polya-szego
  • Theorem 2.4: Schoenberg42
  • ...and 103 more