Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming

Laurent Manivel, Mateusz Michałek, Leonid Monin, Tim Seynnaeve, and Martin Vodička

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Submission date: 02. Dec. 2020
Bibtex
MSC-Numbers: 62R01, 14M17, 14N10
Keywords and phrases: maximum likelihood degree, degree of semidefinite programming, complete quadrics, enumerative geometry, polynomiality
Link to arXiv: See the arXiv entry of this preprint.

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.