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

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.