@inproceedings{2009-C-RepresentationalBregmanVoronoi-ISVD
, author={Frank Nielsen and Richard Nock}
, title={The dual Voronoi diagrams with respect to representational Bregman divergences}
, booktitle={International Symposium on Voronoi Diagrams (ISVD)}
, month={June}
, year={2009}
, address={DTU Lyngby, Denmark}
, publisher={IEEE}
, abstract={
  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 closedform formula for
 their centroids and describe their dual Voronoi diagrams on the induced statistical
 manifolds.
  }
}