# 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.

To view the code, click the link below.

To run the code yourself, download the following Jupyter notebook in Sage `MLdegree.ipynb`

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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