A universality theorem for nonnegative matrix factorizations

Understand

Let $A$ be a matrix with nonnegative real entries.

• A nonnegative factorization of size $k$ is a representation of $A$ as a sum of $k$ nonnegative rank-one matrices.
• The space of all such factorizations is a bounded semialgebraic set, and we prove that spaces arising in this way are universal.
• More presicely, we show that every bounded semialgebraic set $U$ is rationally equivalent to the set of nonnegative size-$k$ factorizations of some matrix $A$ up to a permutation of matrices in the factorization.