Hi,
I want to calculate
$
f_{\mathbf x}(x_1,\ldots,x_k)\, =
\frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}}
\exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})
\right),
$
for a huge ($k$ up 100.000) multivariate Gaussian distribution where $\boldsymbol\Sigma$ is sparse. Cholesky decomposition works fine if the $k$ is smallish and has the advantage that the determinant can be calculated easily from the factors. For larger $k$ I would like to use CG (conjugate gradient optimization), which is really fast to compute $({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})$, however I still need the determinant and have no idea how to (efficiently) compute or approximate it.
Are there any algorithms designed for this problem?
Thank you!
Manuel
