Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

  • Laurent Manivel

    Université de Toulouse, CNRS, France
  • Mateusz Michałek

    Universität Konstanz, Germany
  • Leonid Monin

    University of Bristol, UK
  • Tim Seynnaeve

    Universität Bern, Switzerland
  • Martin Vodička

    Šafárik University, Košice, Slovakia
Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming cover

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Abstract

We establish connections between the maximum likelihood degree (ML-degree) for linear concentration models, the algebraic degree of semidefinite programming (SDP), and Schubert calculus for complete quadrics. We prove a conjecture by Sturmfels and Uhler on the polynomiality of the ML-degree. We also prove a conjecture by Nie, Ranestad and Sturmfels providing an explicit formula for the degree of SDP. The interactions between the three fields shed new light on the asymptotic behaviour of enumerative invariants for the variety of complete quadrics. We also extend these results to spaces of general matrices and of skew-symmetric matrices.

Cite this article

Laurent Manivel, Mateusz Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodička, Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming. J. Eur. Math. Soc. (2023), published online first

DOI 10.4171/JEMS/1330