Maximum Likelihood for Matrices with Rank Constraints

Abstract

Maximum likelihood estimation is a fundamental optimization problem in statistics. Westudy this problem on manifolds of matrices with bounded rank. These represent mixtures of distributionsof two independent discrete random variables. We determine the maximum likelihood degree for a rangeof determinantal varieties, and we apply numerical algebraic geometry to compute all critical points oftheir likelihood functions. This led to the discovery of maximum likelihood duality between matrices ofcomplementary ranks, a result proved subsequently by Draisma and Rodriguez.