Large Structures Seminar: Yaroslav Shitov

Let F be an ordered field and A a nonnegative matrix over F (that is, a matrix whose entries are all nonnegative). The nonnegative rank of A is the smallest k such that A=BC, for some nonnegative matrices B and C of sizes mxk and kxn, respectively. Nonnegative factorizations are important for applications in optimization, statistics, quantum mechanics, and other branches of mathematics. In this talk, I will describe the geometric characterization of nonnegative rank, explain why it is useful in optimization, and discuss some new techniques and results. Namely, I will show that (i) the nonnegative ranks of A with respect to F and some field containing F can be different, and
(ii) every convex n-gon is a projection of a polytope with o(n) facets.