Suppose $X\in \mathbb{R}^{m\times n}, W\in \mathbb{R^{m\times r}_+}, H\in \mathbb{R^{r\times n}_+},$ i.e. $W,H$ are nonnegative matrices and $r\le \min (m,n).$
Are $\displaystyle\arg\min_{W,H} \|X-WH\|_F^2$ and $\displaystyle\arg\min_{W,H} \|X^+-WH \|_F^2$ equivalent if we consider the nontrivial solutions? ($X^+$ are the nonnegative part of the matrix $X, X=X^+ +X^-$. )
In other words, if $(W_0,H_0) \in \arg\min\|X^+-WH\|_F^2,$ then do we have $(W_0,H_0)\in\arg\min \|X-WH \|_F^2 $ ?
Intuitively, I think they are equivalent because $\|X^+ - WH \| \le \|X-WH \|.$ If $X^+ = W_0H_0,$ then it is obvious that the statement is true. However, I did not know how to prove it rigorously for the general case.
If they are not equivalent, can someone give me a counterexample?
Unfortunately, I don't think they are equivalent. As a thought experiment, if it is true that $\displaystyle\arg\min_{W,H} \|X-WH\|_F^2$ and $\displaystyle\arg\min_{W,H} \|X^+-WH \|_F^2$ are equivalent then for all $X_{ij}<0$ it doesn't matter what values they take. If $X_{ij} = -0.1$ or $X_{ij} = -10^{100}$ you would have the same $W_0$ and $H_0$. When trying to minimize the forbenius norm, the term $(X_{ij} - [WH]_{ij})^{2}$ will be the dominate term when $X_{ij}$ is a very large negative number. The effect would be that as $X_{ij}\to -\infty$ that $[WH]_{ij}\to 0$. The only way that $[WH]_{ij}$ is uneffected by scaling $X_{ij}$ is if $[WH]_{ij} = 0$. But in that case $X^+ = WH$. So unless it is always possible to find $W$ and $H$ that satisfy $X^+ = WH$ I don't think it is possible.