Is k-means with Mahalanobis a valid option for clustering?

Question

I want more info into if k-means with Mahalanobis distance is a mathematically/methodologically correct option for datasets with different variance clusters. The steps are:

1. Create aggregate datasets (initially randomly or another way, doesn't matter)
2. Estimate mu, sigma for each aggregate/cluster dataset
3. Recompute clusters by calculating the Mahalanobis distance of each point to each cluster and updating the clusters.
4. go to 2 until clusters do not change.

I have seen this implementation used, also have seen it in dissertations. Yet something doesn't feel comfortable about it. There is no way to have an 'absolute' best clustering (i.e. silhouette metric) with Mahalanobis. You can only estimate the error (e.g. Bayesian Information Criterion) of your model (different normal distributions) on your data. And you can definitely over-fit. Is this still 'correct' in terms of clustering and methodologically accepted? Is this still termed 'k-means' clustering?

I guess must be valid since I haven't seen any argumentation against it. Yet I feel obliged to ask, just to make sure, before I use it.

Thanks a bunch.

Answer 1

It doesnt feel right cause there are convergence problems. See here

It has tendency to fail. With kmeans and euclidian distance you have some really nice mathematical properties and you can gurantee convergence.

Answer 2

It depends on the case that you are going to apply clustering. If your underlying distributions are multivariate gaussians, Mahalanobis distance might be useful. In most cases k-means is combined with Euclidean distance. However, there are cases where Euclidean distance is not useful e.g. text clustering as cosine similarity seems to be the appropriate metric.