Convergence problems of Mahalanobis distance-based k-means clustering

Mahalanobis distance is used for clustering and appears in different scenarios. Sometimes the same covariance is shared for all the clusters. This assumption is very restricted and it might be more meaningful that each cluster will be defined not only by its centroid but also with the covariance matrix. However, its use for k-means algorithm is not appropriate for optimization. It might lead to a good and meaningful clustering, but this is a fact of empirical observation and is not due to the algorithm's convergence. In this study we will show that the overall distance may not decrease from one iteration to another, and that, to ensure convergence, some constraints must be added. Moreover, we will show that in an unconstrained clustering, the cluster covariance matrix is not a solution of the optimization process, but a constraint.