Nonnegative Matrix Factorization Requires Irrationality
  Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix $M$ always has an NMF of minimal inner dimension $d$ whose factors $W$ and $H$ are also rational. We answer this question negatively, by exhibiting a matrix for which $W$ and $H$ require irrational entries. 
Dmitry Chistikov, Stefan Kiefer, Ines Maru\v{s}i\'c, Mahsa   Shirmohammadi, James Worrell
cs.CC cs.LG math.NA