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March 2018 Exponentially concave functions and a new information geometry
Soumik Pal, Ting-Kam Leonard Wong
Ann. Probab. 46(2): 1070-1113 (March 2018). DOI: 10.1214/17-AOP1201

Abstract

A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper, we showed that gradient maps of exponentially concave functions provide solutions to a Monge–Kantorovich optimal transport problem and give a better gradient approximation than those of ordinary concave functions. The approximation error, called L-divergence, is different from the usual Bregman divergence. Using tools of information geometry and optimal transport, we show that L-divergence induces a new information geometry on the simplex consisting of a Riemannian metric and a pair of dually coupled affine connections which defines 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 an interesting application of determining the optimal trading frequency in stochastic portfolio theory.

Citation

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Soumik Pal. Ting-Kam Leonard Wong. "Exponentially concave functions and a new information geometry." Ann. Probab. 46 (2) 1070 - 1113, March 2018. https://doi.org/10.1214/17-AOP1201

Information

Received: 1 June 2016; Revised: 1 January 2017; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864080
MathSciNet: MR3773381
Digital Object Identifier: 10.1214/17-AOP1201

Subjects:
Primary: 60E05
Secondary: 52A41

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 2 • March 2018
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