For a problem of the form $X\approx S$, where $X$ is an $n\times n$ symmetric but not always PD matrix, and $S$ its approximate, I am trying to minimize the log-determinant divergence between $X$ and $S$, that is: $$\min_S \text{Tr}(XS^{-1}) - \log\det(XS^{-1})$$
However, the corresponding algorithm I developed and implemented only works with synthetic data. Real data does not give the expected results.
Does anyone have any idea what the problem is?
I will appreciate any help from you. Thank you.