TY - JOUR
T1 - On conformal divergences and their population minimizers
AU - Nock, Richard
AU - Nielsen, Frank
AU - Amari, Shun Ichi
N1 - Publisher Copyright:
© 2015 IEEE.
PY - 2016/1/1
Y1 - 2016/1/1
N2 - 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 with 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.
AB - 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 with 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.
KW - (u
KW - Ordinary Bregman divergences
KW - Total Bregman divergences
KW - V)-geometric structure
UR - http://www.scopus.com/inward/record.url?scp=84959387247&partnerID=8YFLogxK
U2 - 10.1109/TIT.2015.2448072
DO - 10.1109/TIT.2015.2448072
M3 - Article
SN - 0018-9448
VL - 62
SP - 527
EP - 538
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
IS - 1
M1 - 7163335
ER -