2017/12/10 by Wong, Ting-Kam Leonard
#FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.1712.03610
Divergences, also known as contrast functions, are distance-like quantities defined on manifolds of non-negative or probability measures. Using the duality in optimal transport, we introduce and study the one-parameter family of L(± α)-divergences. It includes the Bregman divergence corresponding to the Euclidean quadratic cost, and the L-divergence introduced by Pal and the author in connection with portfolio theory and a logarithmic cost function. 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 sectional curvatures, and a generalized Pythagorean theorem holds true. 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.