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## Effects of adding loading factors to a covariance matrix

From my previous post, we know that the update equation for covariance matrix might not be numerically stable because of the matrix not being positive definite. An easy way to stabilize the algorithm is to add a relatively small positive number a.k.a. loading factor to the diagonal entries of the covariance matrix. But, Does the factor loading affect the likelihood or the convergence of the EM algorithm?

Apparently, adding the loading factor to the covariance matrix does impact the log-likelihood value. I made some experiments on the issue, and let me share the results with you as seen in the learning curve (log-likelihood curve) of ITSBN with EM algorithm below. The factor is applied to the matrix only when the determinant of the covariance matrix is smaller than $10^{-6}$. There are 5 different factors used in this experiment listed as follows; $10^{-8}, 10^{-6}, 10^{-4}, 10^{-3}, 10^{-2}$. The results show that the learning curves are still monotonically increasing* and level off near the end. Furthermore, we found that the level-off value are highly associated with the value of the factor. The bigger the factor, the smaller the level-off value. This suggested that we should pick smallest value of factor as possible in order to stay as close as the ideal learning curve as possible. Note that the loading factor is not added to the covariance matrix until the second iteration.

log-likelihood curve with different loading factors

* Though I don’t think this is always the case because the factor is not consistently added to the matrix, and hence when it is added, it might pull the log-likelihood up to a low value. However, it is empirically shown that the log-likelihood is still monotonically increasing when the factor is big.

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