Algebraic boundary of matrices of nonnegative rank at most three

Abstract

Understanding the boundary of the set of matrices of nonnegative rank at most $r$ is important for applications in nonconvex optimization. The Zariski closure of the boundary of the set of matrices of nonnegative rank at most $3$ is reducible. We give a minimal generating set for the ideal of each irreducible component. In fact, this generating set is a Gröbner basis with respect to the graded reverse lexicographic order. This solves a conjecture by Robeva, Sturmfels and the last author.