Given $X \in \mathbb{R}^{N_x}$, $Y \in \mathbb{R}^{N_y}$ and $\beta \in \mathbb{R}^+$, so
$W=YX^T(XX^T+\beta I)^{-1}=AB^{-1}$
where $A=YX^T \in \mathbb{R}^{N_y \times N_x}$ and $B=XX^T+\beta I \in \mathbb{R}^{N_x \times N_x}$.
If we select an arbitrary number of indices < $N_x$ by some sort of importance, and so considering only some columns of matrix $A$ and some columns and rows of matrix $B$. The rest of $A$ and $B$ are zeros. Sparsify $A$ and $B$ breaks the inversion and the computation of $W$.
For example, if A is a matrix of $2 \times 5$ and $B$ is a matrix of $5 \times 5$ with $N_x=5$. If we select indexes [2,4], the columns [2,4] of matrix A are non-zeros, and the columns [2,4] and rows [2,4] of matrix B are non-zeros; the rest of elements of matrix $A$ and $B$ are zeros.
Computing $W^*$ from sparse $A$ and $B$ break the solution having low residuals $W^*X-Y$ with respect to $WX-Y$.
There is a way of solve this problem? Is it possible to decompose $B$ in some way in order to invert it more easily?