Article,
On conformal divergences and their population minimizers
(2013)cite arxiv:1311.5125.
Abstract
Total Bregman divergences are a recent tweak of ordinary Bregman divergences
originally motivated by applications that required invariance by rotations.
They have displayed superior results compared to ordinary Bregman divergences
on several clustering, computer vision, medical imaging and machine learning
tasks. These preliminary results raise two important problems : First, report a
complete characterization of the left and right population minimizers for this
class of total Bregman divergences. Second, characterize a principled superset
of total and ordinary Bregman divergences with good clustering properties, from
which one could tailor the choice of a divergence to a particular application.
In this paper, we provide and study one such superset with interesting
geometric features, that we call conformal divergences, and focus on their left
and right population minimizers. Our results are obtained in a recently coined
$(u, v)$-geometric structure that is a generalization of the dually flat affine
connections in information geometry. We characterize both analytically and
geometrically the population minimizers. We prove that conformal divergences
(resp. total Bregman divergences) are essentially exhaustive for their left
(resp. right) population minimizers. We further report new results and extend
previous results on the robustness to outliers of the left and right population
minimizers, and discuss the role of the $(u, v)$-geometric structure in
clustering. Additional results are also given.
