## Tags : Dually flat spaces

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## Mar 14, 2009

### Dually flat spaces and canonical divergences

Well most of us are familiar with Bregman divergences allows one to extend many familiar algorithms such as the celebrated k-means clustering algorithm. Bregman divergences are known to be the canonical divergences of dually flat spaces. Euclidean geometry is simply self-dually flat (induced by the Legendre self-dual paraboloid function).

However if we consider separable divergence defined by means of a monotonous function called the representation funcrton acting on coordinate systems and a convex function, then we can define generalized canonical divergences as explained in Jun Zhang's seminal paper:
Zhang, J. 2004. Divergence function, duality, and convex analysis. Neural Comput. 16, 1 (Jan. 2004), 159-195. DOI= http://dx.doi.org/10.1162/08997660460734047

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p> Now Eguchi and Copas' beta divergences (KL for beta=0) also induce a dually flat structure both on the manifold of positive arrays (and also the submanifold of probability vectors).
S. Eguchi and J. Copas (2002): A class of logistic-type discriminant functions, Biometrika, 89, 1-22. (

So let us keep in mind that dually flat spaces is a notion that encompasses Bregman divergences... A philosophical egg and chicken question of the link between geometry vs distances.