Nonnegative Matrix Factorization Requires Irrationality

The nonnegative rank of a nonnegative matrix M is the smallest number d such that M can be written as the sum of d nonnegative rank-1 matrices. This notion has applications in a variety of areas, including automata theory, communication complexity, document clustering, and recommender systems. A longstanding open problem is whether, when M is a rational matrix, the summands in the above rank decomposition can always be chosen to be rational. In this talk we resolve this problem negatively and discuss consequences of this result for the computational complexity of computing nonnegative rank. This is joint work with Dmitry Chistikov, Stefan Kiefer, Ines Marusic, and Mahsa Shirmohammadi.