K-Means and Gaussian Mixture Models on Lie Groups: Application to Geometrical Clustering

This paper investigates two new clustering algorithms dedicated to Lie groups, adapted from the well known K-Means algorithm and Gaussian mixture models. More precisely, the K-Means algorithm is generalized to handle observations belonging to a Lie group (LG) with an appropriate intrinsic metric. New Gaussian mixture models are investigated for data living in LGs by using concentrated Gaussian distributions. The consistency and the performance of the resulting clustering methods are numerically validated for data lying on the LGSE(2), by comparison with the state of the art for synthetic data, pseudo-real data generated using an ultrasound sensor model and real data from the MNIST dataset.