几何信息

IF 1.8 3区 数学 Q1 MATHEMATICS
Shun-ichi Amari
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引用次数: 0

摘要

信息几何是从研究概率分布族的不变结构中产生的。这种不变性唯一地决定了概率分布流形中的二阶对称张量g和三阶对称张量T。一对这样的张量(g, T)定义了一个黎曼度规和一对仿射连接,它们共同保持了这个度规。信息几何涉及研究具有一对对偶仿射连接的黎曼流形。这种结构也来源于不对称散度函数和仿射微分几何。对偶平坦黎曼流形在各种应用中特别有用,因为广义的勾股定理和投影定理成立。沃瑟斯坦距离给出了概率分布的另一个重要几何形状,它是非不变的,但负责样本空间的度量性质。我试图构建熵正则化Wasserstein距离的信息几何。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Information geometry

Information geometry has emerged from the study of the invariant structure in families of probability distributions. This invariance uniquely determines a second-order symmetric tensor g and third-order symmetric tensor T in a manifold of probability distributions. A pair of these tensors (g, T) defines a Riemannian metric and a pair of affine connections which together preserve the metric. Information geometry involves studying a Riemannian manifold having a pair of dual affine connections. Such a structure also arises from an asymmetric divergence function and affine differential geometry. A dually flat Riemannian manifold is particularly useful for various applications, because a generalized Pythagorean theorem and projection theorem hold. The Wasserstein distance gives another important geometry on probability distributions, which is non-invariant but responsible for the metric properties of a sample space. I attempt to construct information geometry of the entropy-regularized Wasserstein distance.

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来源期刊
CiteScore
3.90
自引率
0.00%
发文量
2
审稿时长
>12 weeks
期刊介绍: The official journal of the Mathematical Society of Japan, the Japanese Journal of Mathematics is devoted to authoritative research survey articles that will promote future progress in mathematics. It encourages advanced and clear expositions, giving new insights on topics of current interest from broad perspectives and/or reviewing all major developments in an important area over many years. An eminent international mathematics journal, the Japanese Journal of Mathematics has been published since 1924. It is an ideal resource for a wide range of mathematicians extending beyond a small circle of specialists. The official journal of the Mathematical Society of Japan. Devoted to authoritative research survey articles that will promote future progress in mathematics. Gives new insight on topics of current interest from broad perspectives and/or reviews all major developments in an important area over many years.
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