I have a symmetric PSD matrix
$$ P = \sum_{i = 1}^N p_ip_i^\top \in \mathbb{R}^{n \times n} $$
where $p_i \in \mathbb{R}^n\ \forall i$. Its SVD is $P = U\Sigma_pU^\top$. I also have another sum
$$A = \sum_{i = 1}^N a_ip_i^\top \in \mathbb{R}^{m \times n}$$
where $a_i \in \mathbb{R}^m\ \forall i$, with SVD $A = V_1\Sigma_aV_2^\top$. There's no relationship between $a_i$ and $p_i$ (actually, in my particular application, all the $a_i$ are random vectors that are independent of $p_j\ \forall j \leq i$).
Obviously, there should be no relationship between $U$ and $V_1$. Given that both $A$ and $P$ have $p_i^\top$ on the right sides of their respective unitary matrices, though, is there a relationship between $U$ and $V_2$? Perhaps even between $\Sigma_p$ and $\Sigma_a$ (where this relationship might need to be expressed in terms of all the $a_i$)?