Linear Gaussian covariance models are Gaussian models with linear constraints on the covariance matrix. Such models arise in many applications, such as stochastic processes from repeated time series data, Brownian motion tree models used for phylogenetic analyses, 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 in fact, with high probability, concave in nature and therefore can be solved using iterative hill-climbing methods.
Caroline Uhler is an Assistant Professor at IST Austria. After completing a master's degree in Mathematics and a bachelor's degree in Biology at the University of Zurich, Caroline received a PhD in Statistics from UC Berkeley in 2011. After postdoctoral appointments at the Institute for Mathematics and its Applications in Minneapolis and at ETH Zurich, Caroline joined IST Austria in 2012. In 2013 she participated in the semester program on Big Data at the Simons Institute at UC Berkeley. Caroline's main research interests are in mathematical statistics (in particular in graphical models, causal inference, and algebraic statistics), in convex optimization, in applied algebraic geometry, and in applications to biology.