The maximum likelihood degree of rank 2 matrices via Euler characteristics

The maximum likelihood degree (ML degree) measures the algebraic complexity of a fundamental computational problem in statistics: maximum likelihood estimation. The Euler characteristic is a classic topological invariant that enjoys many nice properties. In this paper, we use Euler characteristics to prove an outstanding conjecture by Hauenstein, the first author, and Sturmfels; we prove a closed form expression for the ML degree of 3 by n rank 2 matrices. More broadly, we show how these techniques give a recursive expression for the ML degree of m by n rank 2 matrices.