Apologies if this has already been answered; I found some similar posts (here and here) but don't feel they answered the specific question I have. Please feel free to correct any misunderstandings in my question.
As I understand it, the classic k-means clustering algorithm forms clusters by minimizing the Euclidean distance between the observations in each cluster and their corresponding cluster centroids. Spherical k-means (see here) follows a similar procedure, but instead minimizes cosine dissimilarity (i.e. 1 - cosine similarity).
Is there an analogous extension of Gaussian mixture models that clusters observations with cosine dissimilarity instead of Euclidean distance? My understanding with GMMs is that the EM algorithm seeks to find the set of means and covariances (for k Gaussian distributions) that maximizes the likelihood of the observed data. Is it correct then to say that this is implicitly minimizing Euclidean distance because the center of each Gaussian distribution is the mean over a set of points? And if so, is there some way to modify the procedure so that it minimizes cosine dissimilarity instead?
Based on this post, I thought that if I normalized my data before running the GMM I would in effect be maximizing cosine similarity. However, I realize that for normalized vectors, minimizing cosine similarity is equivalent to minimizing the square of Euclidean distance, not the Euclidean distance. The results from running the GMM with normalized data looked similar to the results from spherical k-means, but of course passing this sniff test in no way means what I have done is correct.
