Linear Gaussian covariance models are Gaussian models with linear constraints on the covariance matrix. Such models arise in stochastic processes from repeated time series data, Brownian motion tree models of phylogenetic data and network tomography models used for analyzing connections in the Internet. Maximum likelihood estimation in this class of models leads to a non-convex optimization problem that typically has many local maxima. Using recent results on the asymptotic distribution of the extreme eigenvalues of the Wishart distribution, we prove that maximum likelihood estimation for linear Gaussian covariance models is, with high probability, concave in nature and hence can be solved using iterative hill-climbing methods.
This talk is based on joint work with Piotr Zwiernik (Universitat Pompeu Fabra, Barcelona) and Caroline Uhler (Massachusetts Institute of Technology).