Logarithmic divergences from optimal transport and Rényi geometry
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Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using a duality motivated by optimal transport, we introduce and study a parameterized family of L^(±α)-divergences which includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the L-divergence introduced by Pal and Wong in connection with portfolio theory and a logarithmic cost. They admit natural generalizations of exponential family that are closely related to the α-family and q-exponential family. In particular, the L^(±α)-divergences of the corresponding potential functions are Rényi divergences. Using this unified framework we prove that the induced geometries are dually projectively flat with constant curvatures, and satisfy a generalized Pythagorean theorem. Conversely, we show that if a statistical manifold is dually projectively flat with constant curvature ±α with α > 0, then it is locally induced by an L^(∓α)-divergence. We define in this context a canonical divergence which extends the one for dually flat manifolds.
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