Linear spaces of symmetric matrices with non-maximal maximum likelihood degree

  • Rosa Winter (MPI MiS, Leipzig)
Live Stream


Maximum likelihood estimation is an optimization problem used to fit empirical data to a statistical model. The number of complex critical points to this problem when using generic data is the maximum likelihood degree (ML-degree) of the model. The concentration matrices of certain models form a spectrahedron in the space of symmetric matrices, defined by the intersection of a linear subspace $\mathcal{L}$ with the cone of positive definite matrices. It is known what the ML-degree should be for such models when $\mathcal{L}$ is generic. In this talk I will describe the 'non-generic' linear subspaces, that is, those for which the corresponding model has ML-degree lower than expected. More specifically, for fixed $k$ and $n$, I will describe the geometry of the Zariski closure in the Grassmanian $G$ $(k,($$\substack{n+1\\2}$$))$ of the set of $k$-dimensional linear subspaces of symmetric $n$ $\times$ $n$ matrices that are 'non-generic' in this sense. I will show that this closed set coincides with the set of linear subspaces of symmetric matrices for which strong duality in semi-definite programming fails.

This is joint work with Yuhan Jiang and Kathlén Kohn.


17.03.20 21.02.22

Nonlinear Algebra Seminar Online (NASO)

MPI for Mathematics in the Sciences Live Stream

Katharina Matschke

MPI for Mathematics in the Sciences Contact via Mail