A panorama of positivity

Abstract

This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). We put emphasis on entrywise operations which preserve positivity, in a variety of guises. Techniques from harmonic analysis, function theory, operator theory, statistics, combinatorics, and group representations are invoked. Some partially forgotten classical roots in metric geometry and distance transforms are presented with comments and full bibliographical references. Modern applications to high-dimensional covariance estimation and regularization are included.

arXiv comment83 pages, LaTeX, extended version with additional references. Split into two parts for publication. "Part 1. Dimension free" appears in "Analysis of Operators on Function Spaces (the Serguei Shimorin Memorial Volume)" published by Birkhauser, and "Part 2. Fixed dimension" is to appear in "Complex Analysis and Spectral Theory" of the CRM Proceedings series published by AMS

References in corpus (7)

• Strictly and non-strictly positive definite functions on spheres
• Sphere packing bounds via spherical codes
• An extension of a theorem of Schoenberg to products of spheres
• Preserving positivity for rank-constrained matrices
• Strictly positive definite kernels on a product of circles
• Matrix positivity preservers in fixed dimension. I
• Schoenberg's theorem for real and complex Hilbert spheres revisited

Cited by in corpus (5)

• QFT, EFT and GFT
• On the sign patterns of entrywise positivity preservers in fixed dimension
• Totally positive kernels, Polya frequency functions, and their transforms
• Smooth entrywise positivity preservers, a Horn-Loewner master theorem, and symmetric function identities
• Sharp nonzero lower bounds for the Schur product theorem