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The nonnegative matrix $V = \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right)$ has nonnegative matrix factors $W = \left( \begin{array}{c} 1 \\ 1 \end{array} \right)$ and $H = \left( \begin{array}{cc} 1 & 1 \end{array} \right)$, i.e. $V=WH$. The identity matrix $\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$, on the other hand, has no exact nonnegative factors of smaller dimensionality.

In general, which nonnegative matrices have nonnegative factors, and which do not?

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  • $\begingroup$ What do you mean by "exact" factors? The identity is the product of two transposition matrices, so obviously you exclude these. $\endgroup$ – David Handelman Oct 18 '14 at 23:30
  • $\begingroup$ Sorry to have to ask, but can you please point out to me which matrices multiply to get the identity matrix? The NMF solver I'm using (in R from CRAN) doesn't find factors for identity. $\endgroup$ – rhombidodecahedron Oct 18 '14 at 23:48
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    $\begingroup$ Put the ones in the lower left and upper right. Square it. Voilà, the identity. $\endgroup$ – David Handelman Oct 19 '14 at 0:23
  • $\begingroup$ Oh! I'm sorry, I meant to specify that the condition that the dimensionality of $W$ and $H$ have to be smaller than $V$. I'll edit the post. $\endgroup$ – rhombidodecahedron Oct 19 '14 at 0:38
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Given the notation in the question, I think the actual concept that the OP is looking for is: Nonnegative rank

(Also, googling for "Nonnegative rank" will bring up several useful links).

Finally, this recent paper on Nonnegative matrix factorization covers what seems to be known about the desired (typically low rank) factorizations.

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