My understanding is that solutions from Non-Negative Matrix Factorization (NMF) are not necessarily unique, and rotations can be imposed during the optimization process or after the solutions have been found. This topic is well represented in the Positive Matrix Factorization literature (e.g., in this manuscript), but is scarce from what I can find in the literature of NMF, and popular packages implementing NMF in statistical libraries. Is there a canonical way of exploring rotations in NMF solutions?


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Non-Negative Matrix Factorization approximates an $n \times m$ matrix $V$ by matrix multiplication of an $n \times r$ matrix $W$ and an $r \times m$ matrix $H$, i.e. $$ V \approx W H,$$

under the additional constraint that the elements of $W$ and $H$ must all be non-negative.

You can introduce an $r \times r$ rotation matrix $R$, such that $R^TR = I$.

$$ V \approx W H = (W R^T) (R H) = \tilde{W} \tilde{H}.$$

The estimated matrix did not change but $W$ changed to $\tilde{W}$ and $H$ changed to $\tilde{H}$.

However, there is no guarantee that the elements of $\tilde{W}$ and $\tilde{H}$ are still non-negative. Since NMF has the additional non-negativity constraints, NMF is not invariant to general rotations, but there may exist some rotations that do not affect the non-negativity of the solution. I am not sure if there is more rigorous work on this.

My guess is that any significant rotation will destroy the non-negativity, which is why rotations are not typically explored in NMF.


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