# A panorama of positivity

## TL;DR

This survey explores the concept of positive semi-definiteness across various mathematical and applied contexts, emphasizing positivity-preserving operations and their applications in high-dimensional data analysis.

## Contribution

It provides a comprehensive overview of positivity-preserving techniques, connecting classical theories with modern applications in covariance estimation and regularization.

## Key findings

- Highlights classical and modern positivity-preserving methods
- Connects harmonic analysis, operator theory, and combinatorics
- Includes applications to high-dimensional covariance estimation

## 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.