I would like to find the square root (or Cholesky decomposition) of $$\Lambda = L_kL_k^T+D, $$ where $L_k$ is lower triangular with rank k (only first k columns have non-zero entries), D is positive diagonal.
$\Lambda$ is large, so I'm trying to massage the expression into something I can work with, but I have had no such luck. Ultimately, I just want to whiten Gaussian random variables according to this precision matrix, so if there's a better way to do that than factorizing $\Lambda$, I'm all ears.
One approach is to use a series of $k$ rank-1 updates. In particular, if $x_1,\dots,x_k$ denote the first $k$ columns of $L_k$, then we can write $$ \Lambda = D + x_1x_1^T + \cdots + x_kx_k^T. $$ To begin, note that $D$ has Cholesky decomposition $D = D^{1/2}D^{1/2}$.