A note on the quasiconvex Jensen divergences and the quasiconvex Bregman divergences derived thereof

We first introduce the class of quasiconvex and quasiconcave Jensen divergences which are oriented (asymmetric) distances, and study some of their properties. We then define the quasiconvex Bregman divergences as the limit case of scaled and skewed quasiconvex Jensen divergences, and report a simple closed-form formula. These quasiconvex Bregman divergences between distinct elements have the property to always have one orientation bounded while the other orientation is infinite. We show that these quasiconvex Bregman divergences can also be interpreted as limit cases of generalized skewed Jensen divergences with respect to comparative convexity by using power means. Finally, we illustrate how these quasiconvex Bregman divergences naturally appear as equivalent divergences for the Kullback-Leibler divergences between densities belonging to a same parametric family of distributions with strictly nested supports.