TY - GEN
T1 - Skew Jensen-Bregman Voronoi diagrams
AU - Nielsen, Frank
AU - Nock, Richard
PY - 2011
Y1 - 2011
N2 - A Jensen-Bregman divergence is a distortion measure defined by a Jensen convexity gap induced by a strictly convex functional generator. Jensen-Bregman divergences unify the squared Euclidean and Mahalanobis distances with the celebrated information-theoretic Jensen-Shannon divergence, and can further be skewed to include Bregman divergences in limit cases. We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of divergences. We show that Jensen-Bregman divergences occur in two contexts: (1) when symmetrizing Bregman divergences, and (2) when computing the Bhattacharyya distances of statistical distributions. Since the Bhattacharyya distance of popular parametric exponential family distributions in statistics can be computed equivalently as Jensen-Bregman divergences, these skew Jensen-Bregman Voronoi diagrams allow one to define a novel family of statistical Voronoi diagrams.
AB - A Jensen-Bregman divergence is a distortion measure defined by a Jensen convexity gap induced by a strictly convex functional generator. Jensen-Bregman divergences unify the squared Euclidean and Mahalanobis distances with the celebrated information-theoretic Jensen-Shannon divergence, and can further be skewed to include Bregman divergences in limit cases. We study the geometric properties and combinatorial complexities of both the Voronoi diagrams and the centroidal Voronoi diagrams induced by such as class of divergences. We show that Jensen-Bregman divergences occur in two contexts: (1) when symmetrizing Bregman divergences, and (2) when computing the Bhattacharyya distances of statistical distributions. Since the Bhattacharyya distance of popular parametric exponential family distributions in statistics can be computed equivalently as Jensen-Bregman divergences, these skew Jensen-Bregman Voronoi diagrams allow one to define a novel family of statistical Voronoi diagrams.
KW - Bhattacharyya distance
KW - Bregman divergences
KW - Jensen's inequality
KW - Jensen-Shannon divergence
KW - Jensen-von Neumann divergence
KW - information geometry
UR - http://www.scopus.com/inward/record.url?scp=81755166793&partnerID=8YFLogxK
U2 - 10.1007/978-3-642-25249-5_4
DO - 10.1007/978-3-642-25249-5_4
M3 - Conference contribution
SN - 9783642252488
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 102
EP - 128
BT - Transactions on Computational Science XIV - Special Issue on Voronoi Diagrams and Delaunay Triangulation
T2 - 7th International Symposium on Voronoi Diagrams
Y2 - 28 June 2010 through 30 June 2010
ER -