Pure Maths Colloquium: Kaie Kubjas

One of many definitions gives the rank of an \(m\times n\) matrix \(M\) as the smallest natural number such that M can be factorized as \(AB\), where \(A\) and \(B\) are \(m\times r\) and \(r\times n\) matrices respectively. In many applications, one is interested in factorizations of a particular form. For example, factorizations with nonnegative entries define the nonnegative rank which is a notion that is used in data mining applications, statistics, complexity theory etc. Nonnegative rank has geometric characterizations using nested polytopes. I will give an overview how these nested polytopes are related to characterizations of the set of matrices of given nonnegative rank and uniqueness of nonnegative matrix factorizations.