The Bregman chord divergence

Distances are fundamental primitives whose choice significantly impacts the performances of algorithms in machine learning and signal processing. However selecting the most appropriate distance for a given task is an endeavor. Instead of testing one by one the entries of an ever-expanding dictionary of {\em ad hoc} distances, one rather prefers to consider parametric classes of distances that are exhaustively characterized by axioms derived from first principles. Bregman divergences are such a class. However fine-tuning a Bregman divergence is delicate since it requires to smoothly adjust a functional generator. In this work, we propose an extension of Bregman divergences called the Bregman chord divergences. This new class of distances does not require gradient calculations, uses two scalar parameters that can be easily tailored in applications, and generalizes asymptotically Bregman divergences.Comment: 10 page

Total Jensen divergences: Definition, Properties and k-Means++ Clustering

We present a novel class of divergences induced by a smooth convex function called total Jensen divergences. Those total Jensen divergences are invariant by construction to rotations, a feature yielding regularization of ordinary Jensen divergences by a conformal factor. We analyze the relationships between this novel class of total Jensen divergences and the recently introduced total Bregman divergences. We then proceed by defining the total Jensen centroids as average distortion minimizers, and study their robustness performance to outliers. Finally, we 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.Comment: 27 page