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.
Cite
@article{arxiv.1812.05482,
title = {A panorama of positivity},
author = {Alexander Belton and Dominique Guillot and Apoorva Khare and Mihai Putinar},
journal= {arXiv preprint arXiv:1812.05482},
year = {2019}
}
Comments
83 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