I have a sparse matrix $X$ which is $m \times n$, where $m>n$ and I am trying solve least squares problem $Xa=b$. I need to solve it multiple times for different $b$'s so I apply Cholesky decomposition on $Y=X^TX$, and solve $Ya=X^Tb$.
The problem is I need to add $k$ rows ($k < n$) to $X$ where each additional row contains only one nonzero element. Then $Y$ will be $Y^' = X^TX + W^TW$ where $W$ is $k\times n$ matrix consists of additional rows. Since $W^TW$ is a diagonal matrix another way to show is $Y^' = X^TX + Iv$ where $v$ is a vector $n x 1$ and contains nonzero elements of additional rows.
I do not want to recompute decomposition for each time when multiple rows added, I checked Woodbury matrix identity by considering $Y^' = X^TX + W^TW$ but it is not very useful when $k$ is close to $n$. Is there any special case for $Y^' = X^TX + Iv$ (may be something like Sherman–Morrison formula) ?
Also other suggestions will be appreciated (including pointing to related references, changing decomposition or whole approach etc.), thank you.