Abstract for the talk on 02.02.2021 (17:00 h)

Nonlinear Algebra Seminar Online (NASO)

Rosa Winter (MPI MIS, Leipzig)
Linear spaces of symmetric matrices with non-maximal maximum likelihood degree
See the video of this talk.

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.


18.10.2021, 14:54