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Approximation Algorithms for LowDistortion Embeddings Into LowDimensional Spaces
 in Proceedings of the 16th Annual ACMSIAM Symposium on Discrete Algorithms
, 2005
"... Abstract We present several approximation algorithms for theproblem of embedding metric spaces into a line, and into the twodimensional plane. Among other results, wegive an O(pn)approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, t ..."
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Cited by 28 (9 self)
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, that minimizes the (standard)multiplicative distortion. We give an improved ~ O(n1/3)approximation for the case of metrics generated by unweighted trees. This is the first result of this type. 1 Introduction Embedding distance matrices into geometric spaces(most notably, into lowdimensional spaces) is a
Bounded geometries, fractals, and lowdistortion embeddings
"... The doubling constant of a metric space (X; d) is thesmallest value * such that every ball in X can be covered by * balls of half the radius. The doubling dimension of X isthen defined as dim(X) = log2 *. A metric (or sequence ofmetrics) is called doubling precisely when its doubling dimension is ..."
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Cited by 198 (40 self)
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is bounded. This is a robust class of metric spaceswhich contains many families of metrics that occur in applied settings.We give tight bounds for embedding doubling metrics into (lowdimensional) normed spaces. We consider bothgeneral doubling metrics, as well as more restricted families such as those
Lowdistortion embeddings of general metrics into the line
 In STOC’05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... A lowdistortion embedding between two metric spaces is a mapping which preserves the distances between each pair of points, up to a small factor called distortion. Lowdistortion embeddings have recently found numerous applications in computer science. Most of the known embedding results are ”absol ..."
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Cited by 25 (8 self)
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. For example, if X is a line metric, then even very simple metrics (an npoint star or an npoint cycle) are embeddable into X only with distortion linear in n. Nevertheless, embeddings into the line (or into lowdimensional spaces) are important for many applications. A solution to this issue is to consider
Embedding ultrametrics into lowdimensional spaces
, 2006
"... We study the problem of minimumdistortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Lowdistortion embeddings of ultrametrics into the plane help visual ..."
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Cited by 5 (5 self)
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We study the problem of minimumdistortion embedding of ultrametrics into the plane and higher dimensional spaces. Ultrametrics are a natural class of metrics that frequently occur in applications involving hierarchical clustering. Lowdistortion embeddings of ultrametrics into the plane help
Laplacian eigenmaps and spectral techniques for embedding and clustering.
 Proceeding of Neural Information Processing Systems,
, 2001
"... Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in ..."
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Cited by 668 (7 self)
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Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded
The geometry of graphs and some of its algorithmic applications
 COMBINATORICA
, 1995
"... In this paper we explore some implications of viewing graphs as geometric objects. This approach offers a new perspective on a number of graphtheoretic and algorithmic problems. There are several ways to model graphs geometrically and our main concern here is with geometric representations that res ..."
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Cited by 524 (19 self)
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tensions. 0 For graphs embeddable in lowdimensional spaces with a small distortion, we can find lowdiameter decompositions (in the sense of [4] and [34]). The parameters of the decomposition depend only on the dimension and the distortion and not on the size of the graph. 0 In graphs embedded this way
Laplacian Eigenmaps for Dimensionality Reduction and Data Representation
, 2003
"... One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a lowdimensional manifold embedded in a highdimensional space. Drawing on the correspondenc ..."
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Cited by 1226 (15 self)
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One of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data. We consider the problem of constructing a representation for data lying on a lowdimensional manifold embedded in a highdimensional space. Drawing
LowDistortion Embeddings of Finite Metric Spaces
 in Handbook of Discrete and Computational Geometry
, 2004
"... INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given their diss ..."
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Cited by 66 (1 self)
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INTRODUCTION An npoint metric space (X; D) can be represented by an n n table specifying the distances. Such tables arise in many diverse areas. For example, consider the following scenario in microbiology: X is a collection of bacterial strains, and for every two strains, one is given
Results 1  10
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