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I am comparing the clustering performance of two closely related machine learning methods: K-means and Gaussian Mixture Models (GMM). Part of this research is selecting the best number of clusters K. One of the measures I am using for this purpose is the within-sum of squared distances.

Now, I know that the K-means algorithm monotonically decreases the within-sum of squared distances as the number of clusters K increases and actually finds a minimum if each observation has its own cluster. Given their relatedness, does the same relation hold for GMM? What is the rationale behind the conclusion?

Any thoughts and/or references would be very much appreciated!

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