Let $A\in \mathbb{S}_+^{d,k}$ be a random rank-$k$ PSD matrix, where $\mathbb{S}_+^{d,k}$ is the set of $d$-dimensional PSD matrices with $\text{rank}=k$, $k>1$. $B$ is the rank-$k$ approximation of $A+zz^\top$:
$B=\arg\min_{X,\text{rank}(X)=k}||A+zz^\top-X||_{Frob}$,
where $z\in \mathbb{R}^d$ is a random vector, and I am interested in the rank of $A-B$ (e.g., $\text{rank}(A-B)$).
Intuitively, my feeling is that $\text{rank}(A-B)$ should be almost always larger than $1$. But is there any way to mathematically characterize $\text{rank}(A-B)$? (e.g., something like almost surely, almost everywhere, with probability 1, or something related to measure theory).
Any help (e.g., comments or references) will be appreciated!