Suppose $A\in \mathbb{R}^{m\times n},x\in\mathbb{R}_+^{m\times 1},y\in\mathbb{R}_+^{1\times n}$. $A$ has exactly one negative entry and others are nonnegative.
Consider the problem $(w,h)=\displaystyle\arg\min_{x,y}\|A-xy\|^2_F.$
More generally, if $A$ is a general matrix, i.e., $A$ may have more than one negative enetry, are there a method or formula to find the solution to the above problem.
My thoughts:
If $x,y$ can be any vectors (no nonnegative constrains), then the best approximation is given by SVD, i.e. $xy = u\sigma v^T,$ where $u,v$ are first left and right singular vectors and $\sigma $ are the largest singular values. Also, if $A$ is nonnegative, then the Perror-Frobenius theorem guarantees that $u,v^T $ are nonnegative.
If the only negative entry of $A$ is not very large, then $AA^T$ and $A^TA$ may be still nonnegative, then by the Perron-Frobenius themorem, the first singular vectors $u,v^T$ are still nonnegative. Then they are the solutions to the problem.
However, if the negative entry is very large, which means if the SVD would have negative entries. So, we cannot use it anymore. I was wondering, for an arbitrary $A$ (only one negative entry), how to find the (theoretical optimal) best approximation $xy$?
For the more general problem (more than one negative entry), are there a method or formula to find the best approximation $(x,y)$?
Thanks!