测量下降：一种新的有限度量嵌入方法
Measured descent: A new embedding method for finite metrics
摘要：
我们设计了一种新的嵌入技术，我们称之为测量下降，基于局部分解度量空间，以不同的速度，根据一些概率测度的密度。这为[Bourgain，1985]和[Rao，1999]中构造有限度量的Frechet嵌入的两种主要方法提供了一个精炼和统一的框架。证明了任意n点度量空间（X，d）嵌入Hilbert空间的偏差为O（sqrt{alpha_X log n}），其中alpha_X是X的可分解性的几何估计.作为直接推论，我们得到了一个O（sqrt{（log lambda_X）\log n}）的失真嵌入，其中λ_X是X的加倍常数.由于λ_X \le n，这个结果恢复了Bourgain定理，但是当度量X在某种意义上是“低维”时，得到了改进的界。我们的嵌入对于任意大小的子集都是体积相关的。一个结果是对于所有1个\leq k \leq n存在（k，O（log n））体积相关嵌入，这是最好的可能，并且肯定地回答了U.费奇我们的技术也被用来肯定地回答Y的问题。Rabinovich，证明了任意加权n点平面图嵌入l_ \infty^{O(log n)}的失真为O（1）.时间复杂度为O（log n），时间复杂度为O（log n）。
英文摘要：
We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Frechet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X,d) embeds in Hilbert space with distortion O(sqrt{alpha_X log n}), where alpha_X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O(sqrt{(log lambda_X) \log n}) distortion embedding, where λ_X is the doubling constant of X. Since λ_X\le n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, ``low-dimensional,'' improved bounds are achieved. Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 \leq k \leq n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in l_\infty^{O(log n)} with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)^2).
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