The Dual Voronoi Diagrams with Respect to Representational Bregman Divergences

We present a generalization of Bregman Voronoi diagrams induced by a Bregman divergence acting on a representation function.Bregman divergences are canonical distortion measures of flat spaces induced by strictly convex and differentiable functions, called Bregman generators.Considering a representation function further allows us to conveniently embed the not necessarily flat source space into a dually flat space for which the dual Voronoi diagrams can be derived from an equivalent power affine diagram. We explain the fundamental dualities induced by the pair of Legendre convex conjugates coupled with a pair of conjugate representations.In particular, we show that Amari's celebrated family of $\alpha$-divergences and Eguchi and Copas's $\beta$-divergences are two cases of representational Bregman divergences that are often considered in information geometry. We report closed-form formula for their centroids and describe their dual Voronoi diagrams on the induced statistical manifolds.

[1]  Richard Nock,et al.  Mixed Bregman Clustering with Approximation Guarantees , 2008, ECML/PKDD.

[2]  Shun-ichi Amari,et al.  Methods of information geometry , 2000 .

[3]  Frank Nielsen,et al.  Bregman Voronoi Diagrams , 2007, Discret. Comput. Geom..

[4]  Mariette Yvinec,et al.  Algorithmic geometry , 1998 .

[5]  Frank Nielsen,et al.  On approximating the smallest enclosing Bregman Balls , 2006, SCG '06.

[6]  J. Copas,et al.  A class of logistic‐type discriminant functions , 2002 .

[7]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[8]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[9]  S. Amari.  Integration of Stochastic Models by Minimizing -Divergence , 2007, Neural Computation.

[10]  Mihoko Minami,et al.  Robust Blind Source Separation by Beta Divergence , 2002, Neural Computation.

[11]  P. Vos,et al.  Geometry of f-divergence , 1991 .

[12]  J. Boissonnat,et al.  Algorithmic Geometry: Frontmatter , 1998 .

[13]  L. Bregman.  The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming , 1967 .

[14]  Bernard Chazelle,et al.  An optimal convex hull algorithm in any fixed dimension , 1993, Discret. Comput. Geom..

[15]  Hiroshi Imai,et al.  Voronoi diagram in statistical parametric space by Kullback-Leibler divergence , 1997, SCG '97.

[16]  Frank Nielsen,et al.  Visualizing bregman voronoi diagrams , 2007, SCG '07.

[17]  Richard Nock,et al.  On Bregman Voronoi diagrams , 2007, SODA '07.

[18]  Frank Nielsen,et al.  Sided and Symmetrized Bregman Centroids , 2009, IEEE Transactions on Information Theory.

[19]  F. Nielsen.  An Interactive Tour of Voronoi Diagrams on the GPU , 2007 .

[20]  Frank Nielsen,et al.  On the Smallest Enclosing Information Disk , 2008, CCCG.

[21]  Inderjit S. Dhillon,et al.  Clustering with Bregman Divergences , 2005, J. Mach. Learn. Res..

[22]  David Letscher,et al.  Delaunay triangulations and Voronoi diagrams for Riemannian manifolds , 2000, SCG '00.