A Panorama of Positivity
A PANORAMA OF POSITIVITY ALEXANDER BELTON, DOMINIQUE GUILLOT, APOORVA KHARE, AND MIHAI PUTINAR 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 empha- sis 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 refer- ences. Modern applications to high-dimensional covariance estimation and regularization are included. Contents 1. Introduction 2 2. From metric geometry to matrix positivity 4 2.1. Distance geometry 4 2.2. Spherical distance geometry 6 2.3. Distance transforms 6 2.4. Altering Euclidean distance 8 2.5. Positive definite functions on homogeneous spaces 11 2.6. Connections to harmonic analysis 15 3. Entrywise functions preserving positivity in all dimensions 18 Date: November 13, 2019. 2010 Mathematics Subject Classification. 15-02, 26-02, 15B48, 51F99, 15B05, 05E05, arXiv:1812.05482v3 [math.CA] 12 Nov 2019 44A60, 15A24, 15A15, 15A45, 15A83, 47B35, 05C50, 30E05, 62J10. Key words and phrases. metric geometry, positive semidefinite matrix, Toeplitz ma- trix, Hankel matrix, positive definite function, completely monotone functions, absolutely monotonic functions, entrywise calculus, generalized Vandermonde matrix, Schur polyno- mials, symmetric function identities, totally positive matrices, totally non-negative matri- ces, totally positive completion problem, sample covariance, covariance estimation, hard / soft thresholding, sparsity pattern, critical exponent of a graph, chordal graph, Loewner monotonicity, convexity, and super-additivity.
[Show full text]