Let $(I_k+BB^T)^{-1}$ be given for $B\in \mathbb{R}^{k\times d}$ where $k\ll d$.
Find $(I_k+BDB^T)^{-1}$ when $D\in \mathbb{R}^{d \times d}$ is a diagonal matrix with $D_{ii}=1/(d_{ii}-\alpha)$ for $d_{ii},\alpha\in\mathbb{R}$ so $D^{-1}$ exists.
You may also assume that $BB^T$ and its eigendecomposition $BB^T=U\Sigma U^T$ are given. The particular values of $k$ and $d$ in my problem are $k=64$ and $d=2048$.