any low complexity method to calculate $(\mathbf{I}+\sum_{i}\mathbf{v}_i\mathbf{v}_i^H)^{-1}$

Question

I know that with Sherman–Morrison formula, we can simply calculate $(\mathbf{I}+\mathbf{v}\mathbf{v}^H)^{-1}$, where $\mathbf{v}$ is a column vetor. What about the case of the summation, i.e., $(\mathbf{I}+\sum_{i}\mathbf{v}_i\mathbf{v}_i^H)^{-1}$

Answer 1

In general $Z=\sum_{i=1}^r u_iv_i^*$ has rank $r$. When $r/n$ is small, using the Woodbury formula has a low complexity. This requires finding $U,C,V$ s.t. $Z=UCV$ where $U\in M_{n,r},C\in GL_r,V\in M_{r,n}$.

I do not think that the hypothesis $u_i=v_i$ greatly reduces the complexity.