Complete quadrics: Schubert calculus for Gaussian models and semidefinite programming
The maximum likelihood degree \(\phi(n,d)\) of the generic linear concentration model is a quantity that is of interest for algbraic statistics, but also admits natural geometric definition: it the the degree of the variety \(\phi(n,d)\) obtained by inverting all matrices in a general \(d\) -dimensional linear space of symmetric \(n \times n\) matrices. Using the geometry of complete quadrics, we prove a conjecture of Sturmfels and Uhler, stating that for fixed \(d\), \(\phi(n,d)\) is a polynomial in \(n\), of degree \(d-1\). Our proof method yields an explicit algorithm for computing these polynomials.
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Project page created: 12/11/2020
Project contributors: Laurent Manivel, Mateusz Michałek, Leonid Monin, Tim Seynnaeve, Martin Vodicka
Code written by: Tim Seynnaeve
Jupyter notebook written by: Tim Seynnaeve
Software used: Sage (Version 9.0)
Corresponding author of this page: Tim Seynnaeve, tim.seynnaeve@kuleuven.be