For matrices $X,Y \in \mathbb{R}^{m \times r},m>r$, let
$$d(X,Y) = \min_{R \in \mathbb{R}^{r \times r}, \, R^\top R=I} \|X-YR\|_2^2$$
Given $Y$, if the $r$-th singular value of $Y$ satisfies $\sigma_r(Y)>0$ and $X$ satisfies $d(X,Y)>c_1 \sigma_r^2(Y)$ for a constant $c_1>0$, then can I bound $\| XX^\top - YY^\top \|_2^2 \geq c_2 \cdot poly(\sigma_r(Y))$ for some $c_2>0$ ?
