There is a way to run a faster $k$-means by using Elkan's method, which uses the triangle inequality to avoid some calculations.
I am trying to think of a way you could do a similar sort of thing for expectation maximization, specifically if you are using Gaussian mixture models.
Unfortunately, I only see weights and probabilities inside the EM algorithm, not anything that's a distance metric.
My best idea was using $k$-means with triangle inequality, run until convergence, and use the final answers as starting cluster centers for EM. But this method doesn't actually do triangle inequality inside EM itself.
Is there a way to use soft clustering and still use triangle inequality?