Nonnegative Matrix Factorization Requires Irrationality

TL;DR

This paper proves that some nonnegative matrices require irrational factors in their minimal nonnegative matrix factorizations, resolving a long-standing open question in the field.

Contribution

It demonstrates that not all minimal NMFs can be rational, providing a counterexample that answers a 1993 open problem.

Findings

Certain matrices have inherently irrational NMF factors

Rational matrices can require irrational solutions in minimal NMF

Addresses a 30-year-old open question in matrix factorization theory

Abstract

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.