Exponentially concave functions and a new information geometry

Abstract. A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. It is known that gradient maps of exponentially concave functions are solutions of a MongeKantorovich optimal transport problem and allow for a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using the tools of information geometry and optimal transport, we show that Ldivergence induces a new information geometry on the simplex. This geometric structure consists of a Riemannian metric and a pair of dually coupled affine connections defining two kinds of geodesics. We show that the induced geometry is dually projectively flat but not flat. Nevertheless, we prove an analogue of the celebrated generalized Pythagorean theorem from classical information geometry. On the other hand, we consider displacement interpolation under a Lagrangian integral action that is consistent with the optimal transport problem and show that the action minimizing curves are dual geodesics. The Pythagorean theorem is also shown to have a remarkable application of determining the optimal frequency of rebalancing in stochastic portfolio theory.