狄利克雷多项式与熵
Dirichlet polynomials and entropy
摘要：
在一个变量${\mathcal{y}}$中的狄利克雷多项式$d$对于某些$n,a_0,\ldots,a_n\in\mathbb{N}$是形式为$d({\mathcal{y}})=a_n n^{\mathcal{y}}+\cdots+a_22^{\mathcal{y}}+a_11^{\mathcal{y}}+a_00^{\mathcal{y}}$的函数。我们将展示如何将狄利克雷多项式视为集合论丛，从而视为经验分布。然后，我们可以考虑相应概率分布的香农熵$H(d)$，并通过$L(d)=2^{H(d)}$定义其长度（或经典的困惑度）。另一方面，我们将定义一个从狄利克雷多项式rig到所谓矩形rig的rig同态$h\colon\mathsf{Dir}\to\mathsf{Rect}$，它的基础集是$\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}$，它的加法结构涉及加权几何平均;我们写为$h(d)=(A(d),W(d))$，并分别称这两个分量为面积和宽度。本文的主要结果是：矩形面积公式$A(d)=L(d)W(d)$对任意Dirichlet多项式$d$成立。换句话说，经验分布的熵可以完全根据应用于其相应狄利克雷多项式的同态$h$来计算。我们还表明，类似的结果举行的交叉熵。
英文摘要：
A Dirichlet polynomial $d$ in one variable ${\mathcal{y}}$ is a function of the form $d({\mathcal{y}})=a_n n^{\mathcal{y}}+\cdots+a_22^{\mathcal{y}}+a_11^{\mathcal{y}}+a_00^{\mathcal{y}}$ for some $n,a_0,\ldots,a_n\in\mathbb{N}$. We will show how to think of a Dirichlet polynomial as a set-theoretic bundle, and thus as an empirical distribution. We can then consider the Shannon entropy $H(d)$ of the corresponding probability distribution, and we define its length (or, classically, its perplexity) by $L(d)=2^{H(d)}$. On the other hand, we will define a rig homomorphism $h\colon\mathsf{Dir}\to\mathsf{Rect}$ from the rig of Dirichlet polynomials to the so-called rectangle rig, whose underlying set is $\mathbb{R}_{\geq0}\times\mathbb{R}_{\geq0}$ and whose additive structure involves the weighted geometric mean; we write $h(d)=(A(d),W(d))$, and call the two components area and width (respectively). The main result of this paper is the following: the rectangle-area formula $A(d)=L(d)W(d)$ holds for any Dirichlet polynomial $d$. In other words, the entropy of an empirical distribution can be calculated entirely in terms of the homomorphism $h$ applied to its corresponding Dirichlet polynomial. We also show that similar results hold for the cross entropy.
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