Let $A = U_A \Sigma_A V_A^\top$ and $B = U_B \Sigma_B V_B^\top$, and $A+B = U \Sigma V^\top$ be the respective singular vector decompositions. Is there some known relationship of the form
$$\| U_A V_A^\top - UV^\top \| \leq \varepsilon,$$
where the norm can be anything sensible (e.g., unitary norms like $\ell_2$ or Frobenius).
What is the dependence of $\varepsilon$ on the spectra of $A$ and $B$?
I understand that the Davis-Kahan theorem answers this to some extent, where $B$ is understood as a perturbation term and it requires $A$ to have some eigen-value separation (or gap).
