Total Jensen divergences: Definition, properties and clustering

We present a novel class of divergences induced by a smooth convex function called total Jensen divergences that are invariant by construction to rotations, a feature inducing a conformal factor on ordinary Jensen divergences. We analyze the relationships between this novel class of total Jensen divergences and the total Bregman divergences. We then define total Jensen centroids, analyze their robustness, and prove that the k-means++ initialization that bypasses explicit centroid computations is good enough in practice to guarantee probabilistically a constant approximation factor to the optimal k-means clustering.