度量空间的最大熵

IF 0.6 4区 数学 Q3 MATHEMATICS
Tom Leinster;Emily Roff
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引用次数: 10

摘要

我们定义了一个单参数熵族,每个熵族为紧致度量空间(或者更一般地说,具有点间相似性概念的紧致Hausdorff空间)上的任何概率测度赋一个实数。这些概括了信息论的香农熵和雷姆尼熵。我们证明了在任意空间X上,存在一个概率度量同时最大化所有这些熵。而且,所有的熵都有相同的最大值:X的最大熵。随着X的增大,最大熵增大,它的渐近性决定了X的几何信息,包括体积和维数。最大化测度本身的大尺度极限提供了一个问题的答案:度量空间上的标准测度是什么?首先,我们研究的不是熵本身,而是它的指数,它的有限形式已经被用来衡量生物多样性。我们的主要定理首先是由伦斯特和梅克斯在有限情况下证明的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The Maximum Entropy of a Metric Space
We define a one-parameter family of entropies, each assigning a real number to any probability measure on a compact metric space (or, more generally, a compact Hausdorff space with a notion of similarity between points). These entropies generalise the Shannon and Renyi entropies of information theory. We prove that on any space X, there is a single probability measure maximising all these entropies simultaneously. Moreover, all the entropies have the same maximum value: the maximum entropy of X. As X is scaled up, the maximum entropy grows; its asymptotics determine geometric information about X, including the volume and dimension. We also study the large-scale limit of the maximising measure itself, arguing that it should be regarded as the canonical or uniform measure on X. Primarily we work not with entropy itself but its exponential, called diversity and (in its finite form) used as a measure of biodiversity. Our main theorem was first proved in the finite case by Leinster and Meckes.
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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