Been thinking through fitting a kind of Gaussian mixture model in more of a neural network style (kind of similar to RNADE or RMADE by Larochelle, without going into details) and see that this could scale really nicely to very large dimensions if I could avoid having to know both $\Sigma$ AND $\Sigma^{-1}$ per the multivariate Gaussian equation. I.e., from what I'm seeing, the Gaussian PDF...

$det(2\pi\Sigma)^{-\frac12}e^{-\frac12(x-\mu)^t \Sigma^{-1}(x-\mu)}$

seems to require both the covariance and precision matrices. I bet the answer is no, but is there a way I could fit parameters for the precision matrix and avoid having to invert it to get the normalization constant (e.g., if i could swap the order of the determinant and inverse sqrt operation, and just need the $\Sigma^{-\frac12}$, I could do a Cholesky decomposition on the precision matrix).

  • 3
    $\begingroup$ Are you looking for $det(A^{-1})=det(A)^{-1}$? $\endgroup$
    – Elvis
    Feb 1, 2018 at 7:50
  • $\begingroup$ ya so that is a legit property? Like the transpose of the inverse is the inverse of the transpose? $\endgroup$
    – JPJ
    Feb 1, 2018 at 20:02
  • $\begingroup$ Well yes. More generally det(AB) = det(A) det(B). It's very surprising that you know very advanced stuff like Cholesky decomposition but you have doubts about this fact! $\endgroup$
    – Elvis
    Feb 1, 2018 at 20:10
  • 1
    $\begingroup$ Well....guess this is the first time I've needed to know it :-) $\endgroup$
    – JPJ
    Feb 1, 2018 at 20:28


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