What functions, when applied to the pairwise Manhattan distances between any $n$ points, result in the Manhattan distances between another set of $n$ points? In this paper, we show that a function has this property if and only if it is Bernstein. This class of functions admits several classical analytic characterizations and includes $f(x) = x^s$ for $0 \leq s \leq 1$ as well as $f(x) = 1-e^{-xt}$ for any $t \geq 0$. While it was previously known that Bernstein functions had this property, it was not known that these were the only such functions. Our results are a natural extension of the work of Schoenberg from 1938, who addressed this question for Euclidean distances. Schoenberg's work has been applied in probability theory, harmonic analysis, machine learning, theoretical computer science, and more. We additionally show that if and only if $f$ is completely monotone, there exists \mbox{$F:\ell_1 \rightarrow \mathbb{R}^n$} for any $x_1, \ldots x_n \in \ell_1$ such that $f(\|x_i - x_j\|_1) = \langle F(x_i), F(x_j) \rangle$. Previously, it was known that completely monotone functions had this property, but it was not known they were the only such functions. The same result but with negative type distances instead of $\ell_1$ is the foundation of all kernel methods in machine learning, and was proven by Schoenberg in 1942.

中文翻译：

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保留曼哈顿距离的功能

应用于任何$ n $点之间的成对曼哈顿距离时，什么函数会导致另一组$ n $点之间的曼哈顿距离？在本文中，我们证明了当且仅当函数是伯恩斯坦时，函数才具有此属性。此类函数接受几种经典的分析特征，包括$ f（x）= x ^ s $表示$ 0 \ leq s \ leq 1 $以及$ f（x）= 1-e ^ {-xt} $ $ t \ geq 0 $。尽管以前知道Bernstein函数具有此属性，但不知道这些是唯一的此类函数。我们的结果是Schoenberg自1938年以来的工作的自然延伸，他针对欧几里得距离解决了这个问题。Schoenberg的工作已应用于概率论，谐波分析，机器学习，理论计算机科学等领域。我们还表明，当且仅当$ f $完全单调时，对于任何$ x_1，\ ldots x_n \ in \ ell_1 $这样的情况，都存在\ mbox {$ F：\ ell_1 \ rightarrow \ mathbb {R} ^ n $} $ f（\ | x_i-x_j \ | _1）= \ langle F（x_i），F（x_j）\ rangle $。以前，已知完全单调函数具有此属性，但不知道它们是唯一的此类函数。相同的结果是负的类型距离而不是$ \ ell_1 $，这是机器学习中所有内核方法的基础，并在1942年由Schoenberg证明。