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Analysis of Operators on Function Spaces pp 117–165Cite as

Birkhäuser

A Panorama of Positivity. I: Dimension Free

Part of the Trends in Mathematics book series (TM)

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.

Keywords

  • Metric geometry
  • Positive semidefinite matrix
  • Toeplitz matrix
  • Hankel matrix
  • Positive definite function
  • Completely monotone functions
  • Absolutely monotonic functions
  • Entrywise calculus
  • Generalized Vandermonde matrix
  • Schur polynomials
  • Symmetric function identities
  • Totally positive matrices
  • Totally non-negative matrices
  • Totally positive completion problem
  • Sample covariance
  • Covariance estimation
  • Hard/soft thresholding
  • Sparsity pattern
  • Critical exponent of a graph
  • Chordal graph
  • Loewner monotonicity
  • Convexity
  • Super-additivity

2010 Mathematics Subject Classification

  • 15-02
  • 26-02
  • 15B48
  • 51F99
  • 15B05
  • 05E05
  • 44A60
  • 15A24
  • 15A15
  • 15A45
  • 15A83
  • 47B35
  • 05C50
  • 30E05
  • 62J10

Serguei Shimorin, in memoriam

D.G. is partially supported by a University of Delaware Research Foundation grant, by a Simons Foundation collaboration grant for mathematicians, and by a University of Delaware Research Foundation Strategic Initiative grant. A.K. is partially supported by Ramanujan Fellowship SB/S2/RJN-121/2017 and MATRICS grant MTR/2017/000295 from SERB (Govt. of India), by grant F.510/25/CAS-II/2018(SAP-I) from UGC (Govt. of India), and by a Young Investigator Award from the Infosys Foundation.

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Notes

  1. 1.

    An alternate proof of sufficiency is to note that \(A := [ \cos \rho ( x_j, x_k ) ]_{j, k = 1}^n\) is a Gram matrix of rank r, hence equal to B T B for some r × n matrix B with unit columns. Denoting these columns by b 1, …, b n ∈ S r−1, the map x j↦b j is an isometry since ρ(x j, x k) and \(\sphericalangle ( y_j, y_k ) \in [ 0, \pi ]\). Moreover, since A has rank r, the b j cannot all lie in a smaller-dimensional sphere.

  2. 2.

    Recall [95] that a metric space (X, ρ) is n-point homogeneous if, given finite sets X 1, X 2 ⊂ X of equal size no more than n, every isometry from X 1 to X 2 extends to a self-isometry of X. This property was first considered by Birkhoff [15], and of course differs from the more common usage of the terminology of a homogeneous space G∕H, whose study by Bochner was mentioned above.

  3. 3.

    This is connected to semi-algebraic geometry and to Hilbert’s seventeenth problem: recall the famous result of Motzkin that there are non-negative polynomials on \(\mathbb {R}^d\) that are not sums of squares, such as x 4 y 2 + x 2 y 4 − 3x 2 y 2 + 1. Such phenomena have been studied in several settings, including polytopes (by Farkas, Handelman, and Pólya) and more general semi-algebraic sets (by Putinar, Schmüdgen, Stengel, Vasilescu, and others).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Lancaster University, Lancaster, UK

    Alexander Belton

  2. University of Delaware, Newark, DE, USA

    Dominique Guillot

  3. Indian Institute of Science, Analysis and Probability Research Group, Bengaluru, India

    Apoorva Khare

  4. University of California at Santa Barbara, Santa Barbara, CA, USA

    Mihai Putinar

  5. Newcastle University, Newcastle upon Tyne, UK

    Mihai Putinar

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  1. Alexander Belton
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  1. Mathematics Department, Lund University, Lund, Sweden

    Prof. Alexandru Aleman

  2. Department of Mathematics, The Royal Institute of Technology, Stockholm, Sweden

    Haakan Hedenmalm

  3. Department of Mathematics, University of South California, Tampa, FL, USA

    Prof. Dmitry Khavinson

  4. Mathematics Department, University of California, Santa Barbara, CA, USA

    Mihai Putinar

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Belton, A., Guillot, D., Khare, A., Putinar, M. (2019). A Panorama of Positivity. I: Dimension Free. In: Aleman, A., Hedenmalm, H., Khavinson, D., Putinar, M. (eds) Analysis of Operators on Function Spaces. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-14640-5_5

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