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Projection Theorems for the Rényi Divergence on $α$-Convex Sets

This paper studies forward and reverse projections for the Rényi divergence of order $α\in (0, \infty)$ on $α$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {\em et al.} in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on $α$-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of forward projection is proved for probability measures on a general alphabet. For $α\in (1, \infty)$, the proof relies on a new Apollonius theorem for the Hellinger divergence, and for $α\in (0,1)$, the proof relies on the Banach-Alaoglu theorem from functional analysis. Further projection results are then obtained in the finite alphabet setting. These include a projection theorem on a specific $α$-convex set, which is termed an {\em $α$-linear family}, generalizing a result by Csiszár for $α\neq 1$. The solution to this problem yields a parametric family of probability measures which turns out to be an extension of the exponential family, and it is termed an {\em $α$-exponential family}. An orthogonality relationship between the $α$-exponential and $α$-linear families is established, and it is used to turn the reverse projection on an $α$-exponential family into a forward projection on a $α$-linear family. This paper also proves a convergence result of an iterative procedure used to calculate the forward projection on an intersection of a finite number of $α$-linear families.