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A universality theorem for nonnegative matrix factorizations
Let be a matrix with nonnegative real entries. A nonnegative
factorization of size is a representation of as a sum of
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 is
rationally equivalent to the set of nonnegative size- factorizations of some
matrix up to a permutation of matrices in the factorization. We prove that,
if is given as the zero locus of a polynomial with
coefficients in , then such a pair can be computed in
polynomial time. This result gives a complete description of the algorithmic
complexity of nonnegative rank, and it also allows one to solve the problem of
Cohen and Rothblum on nonnegative factorizations restricted to matrices over
different subfields of .Comment: 8 pages, an introduction is added, the proofs are written more
accuratel