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Author Chan, T-H Hubert ♦ Gupta, Anupam ♦ Talwar, Kunal
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2010
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Euclidean embedding ♦ Metric spaces ♦ Dimension reduction
Abstract We consider the problem of embedding a metric into low-dimensional Euclidean space. The classical theorems of Bourgain, and of Johnson and Lindenstrauss say that any metric on $\textit{n}$ points embeds into an $\textit{O}(log$ $\textit{n})-dimensional$ Euclidean space with $\textit{O}(log$ $\textit{n})$ distortion. Moreover, a simple “volume” argument shows that this bound is nearly tight: a uniform metric on $\textit{n}$ points requires nearly logarithmic number of dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to low-dimensional embeddings. In other words, do $\textit{doubling}$ metrics, that do not have large uniform submetrics, and thus no volume hurdles to low dimensional embeddings, embed in low dimensional Euclidean spaces with small distortion? In this article, we give a positive answer to this question. We show how to embed any doubling metrics into $\textit{O}(log$ log $\textit{n})$ dimensions with $\textit{O}(log$ $\textit{n})$ distortion. This is the first embedding for doubling metrics into fewer than logarithmic number of dimensions, even allowing for logarithmic distortion. This result is one extreme point of our general trade-off between distortion and dimension: given an $\textit{n}-point$ metric $\textit{(V,d)}$ with doubling dimension $dim_{D},$ and any target dimension $\textit{T}$ in the range $Ω(dim_{D}$ log log $\textit{n})$ ≤ $\textit{T}$ ≤ $\textit{O}(log$ $\textit{n}),$ we show that the metric embeds into Euclidean space $R^{T}$ with $\textit{O}(log$ $\textit{n}$ &sqrt; $dim_{D}/T)$ distortion.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2010-05-03
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 57
Issue Number 4
Page Count 26
Starting Page 1
Ending Page 26

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Source: ACM Digital Library