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A Panorama of Positivity. I: Dimension Free

Belton, Alexander and Guillot, Dominique and Khare, Apoorva and Putinar, Mihai (2019) A Panorama of Positivity. I: Dimension Free. [Book Chapter]

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Official URL: https://doi.org/10.1007/978-3-030-14640-5_5


Abstract This survey contains a selection of topics unified by the concept of positive semidefiniteness (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.

Item Type: Book Chapter
Publisher: Springer International Publishing
Additional Information: The copyright of this article belongs to the Author(s).
Keywords: Absolutely monotonic functions; Chordal graph; Completely monotone functions; Convexity; Covariance estimation; Critical exponent of a graph; Entrywise calculus; Generalized Vandermonde matrix; Hankel matrix; Hard/soft thresholding; Loewner monotonicity; Metric geometry; Positive definite function; Positive semidefinite matrix; Sample covariance; Schur polynomials; Sparsity pattern; Super-additivity; Symmetric function identities; Toeplitz matrix; Totally non-negative matrices; Totally positive completion problem; Totally positive matrices
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 21 Oct 2022 10:46
Last Modified: 21 Oct 2022 10:46
URI: https://eprints.iisc.ac.in/id/eprint/77694

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