1 research outputs found
A note on the quasiconvex Jensen divergences and the quasiconvex Bregman divergences derived thereof
We first introduce the class of strictly quasiconvex and strictly
quasiconcave Jensen divergences which are oriented (asymmetric) distances, and
study some of their properties. We then define the strictly quasiconvex Bregman
divergences as the limit case of scaled and skewed quasiconvex Jensen
divergences, and report a simple closed-form formula which shows that these
divergences are only pseudo-divergences at countably many inflection points of
the generators. To remedy this problem, we propose the -averaged
quasiconvex Bregman divergences which integrate the pseudo-divergences over a
small neighborhood in order obtain a proper divergence. The formula of
-averaged quasiconvex Bregman divergences extend even to
non-differentiable strictly quasiconvex generators. These quasiconvex Bregman
divergences between distinct elements have the property to always have one
orientation finite 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 probability densities belonging to a same parametric family
of distributions with nested supports.Comment: 19 pages, 3 figures, 1 tabl