有限体积空间与稀疏化
Finite Volume Spaces and Sparsification
摘要：
我们介绍和研究有限$d$ -体积-有限度量空间的高维推广。开发了一个合适的组合机械，我们定义了$\ell_1$ -卷，并表明它们包含欧几里德卷和超树卷。我们证明了它们可以近似任何具有$O(n^d)$乘性畸变的$d$ -体积。另一方面，与Bourgain的$d=1$定理相反，在$n$顶点上存在一个$2$ -体积，它不能被任何失真小于$\tildeΩ(n^{1/5})$的$\ell_1$ -体积近似。我们进一步解决的问题，$\ell_1$ -降维的上下文中的$\ell_1$卷，并表明，这种现象确实发生，虽然不是同样惊人的程度，因为它为欧几里德度量和卷。特别是，我们表明，任何$\ell_1$度量$n$点上可以$(1+ ε)$ -近似的$O(n/ε^2)$切割度量的总和，提高了以前最好的已知界的$O(n \log n)$由于Schechtman。为了处理降维问题，我们扩展了Karger和Bencz{ú}r以及Spielman等人提出的技术和思想。在图稀疏化的背景下，并开发具有广泛应用的通用方法。
英文摘要：
We introduce and study finite $d$-volumes - the high dimensional generalization of finite metric spaces. Having developed a suitable combinatorial machinery, we define $\ell_1$-volumes and show that they contain Euclidean volumes and hypertree volumes. We show that they can approximate any $d$-volume with $O(n^d)$ multiplicative distortion. On the other hand, contrary to Bourgain's theorem for $d=1$, there exists a $2$-volume that on $n$ vertices that cannot be approximated by any $\ell_1$-volume with distortion smaller than $\tildeΩ(n^{1/5})$. We further address the problem of $\ell_1$-dimension reduction in the context of $\ell_1$ volumes, and show that this phenomenon does occur, although not to the same striking degree as it does for Euclidean metrics and volumes. In particular, we show that any $\ell_1$ metric on $n$ points can be $(1+ ε)$-approximated by a sum of $O(n/ε^2)$ cut metrics, improving over the best previously known bound of $O(n \log n)$ due to Schechtman. In order to deal with dimension reduction, we extend the techniques and ideas introduced by Karger and Bencz{ú}r, and Spielman et al.~in the context of graph Sparsification, and develop general methods with a wide range of applications.
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