英文标题：
《Exponentially concave functions and a new information geometry》
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作者：
Soumik Pal, Ting-Kam Leonard Wong
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最新提交年份：
2017
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英文摘要：
  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.
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中文摘要：
如果一个函数的指数是凹的，那么它就是指数凹的。我们考虑单位单纯形上的指数凹函数。在前一篇文章中，我们证明了指数凹函数的梯度映射提供了Monge-Kantorovich最优运输问题的解，并且给出了比普通凹函数更好的梯度近似。近似误差称为L-散度，与通常的布雷格曼散度不同。利用信息几何和最优传输工具，我们证明了L-散度在由一个黎曼度量和一对定义两类测地线的双耦合仿射连接组成的单纯形上导出了一个新的信息几何。我们证明了诱导几何是对偶投影平坦的，但不是平坦的。然而，我们证明了经典信息几何中著名的广义勾股定理的类似物。另一方面，我们考虑了与最优运输问题一致的拉格朗日积分作用下的位移插值，并证明了作用最小化曲线是双测地线。毕达哥拉斯定理在确定随机投资组合理论中的最佳交易频率方面也有着有趣的应用。
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分类信息：

一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Computer Science        计算机科学
二级分类：Information Theory        信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Mathematics        数学
二级分类：Information Theory        信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
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