The type of result I want is: given matrix $A\in \mathbb{R}^{m\times n}$ and error tolerance $\epsilon$, what is a lower bound on $k$ such that $\|A - UV\|_{??}\le \epsilon$, where $U \in\mathbb{R}^{m\times k}$, $V\in \mathbb{R}^{k\times n}$, and the norm is an arbitrary matrix norm.
I'm wondering if such a result exists, and if it does, a citation would be helpful.
