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
Alternatively, you can run the code online through Binder without installing any software:
Project page created: 12/11/2020
Project contributors: Laurent Manivel, Mateusz Michalek, 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, email@example.com