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We provide an example of a $21\times 21$ matrix with nonnegative integer entries which can be written as a sum of $19$ nonnegative rank-one matrices but not as a sum of $19$ rational nonnegative rank-one matrices.
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S. A. Vavasis, On the complexity of nonnegative matrix factorization, SIAM Journal on Optimization
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T. Rothvoss, Some 0/1 polytopes need exponential size extended formulations, Mathematical Programming
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Cited alongside, same era.
K. Kubjas, E. Robeva, B. Sturmfels, Fixed points of the EM algorithm and nonnegative rank boundaries, Annals of Statistics
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Cited alongside, same era.
Y. Shitov, Nonnegative rank depends on the field, preprint (2015) arXiv:1505.01893
2015
Later among the works it cites.
J. Gouveia, H. Fawzi, R. Z. Robinson, Rational and real positive semidefinite rank can be different, Operations Research Letters
2016
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