Measured Descent: A New Embedding Method for Finite Metrics

Krauthgamer, Robert; Lee, J.R.; Mendel, Manor; Naor, Assaf

doi:10.1109/focs.2004.41

Cited by 86 publications

(115 citation statements)

References 29 publications

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“…In the off-line setting, Krauthgamer et al [15] have obtained embeddings of doubling metrics into Hilbert space with distortion O( √ log n). Their approach however is based on the random partitioning scheme of Calinescu, Karloff, and Rabani [5], and it is not known how to perform this step in the on-line setting.…”

Section: Theoremmentioning

confidence: 99%

Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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We initiate the study of on-line metric embeddings. In such an embedding we are given a sequence of n points X = x 1 , . . . , x n one by one, from a metric space M = (X, D). Our goal is to compute a low-distortion embedding of M into some host space, which has to be constructed in an on-line fashion, so that the image of each x i depends only on x 1 , . . . , x i . We prove several results translating existing embeddings to the on-line setting, for the case of embedding into p spaces, and into distributions over ultrametrics.

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Section: Theoremmentioning

confidence: 99%

Online Embeddings

Indyk

Magen

Sidiropoulos

et al. 2010

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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“…ε ∈ ∆ im (C)) we shall denote by E C (ε) the set of edges of C which ε inserted to S C . Denoting D ε = d G k (ε, ε C ) + 1 we see that (29) implies that for e ∈ E C (ε) we have…”

Section: -Cycle G (K)mentioning

confidence: 99%

“…The following fundamental result is due to [19] when µ is the uniform measure on X. The case of general measures was observed in [29,32], and the speci c numerical constants used below are taken from [34].…”

mentioning

confidence: 99%

Maximum gradient embeddings and monotone clustering

Mendel

2010

Combinatorica

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Let (X, d X ) be an n-point metric space. We show that there exists a distribution D over non-contractive embeddings into trees f : X → T such that for every x ∈ X,where C is a universal constant. Conversely we show that the above quadratic dependence on log n cannot be improved in general. Such embeddings, which we call maximum gradient embeddings, yield a framework for the design of approximation algorithms for a wide range of clustering problems with monotone costs, including fault-tolerant versions of k-median and facility location. IntroductionMetric embeddings are an invaluable tool in analysis, Riemannian geometry, group theory, graph theory, and the design of approximation algorithms. In most cases embeddings are used to simplify a geometric object that we wish to understand, or on which we need to preform certain algorithmic tasks. Thus one tries to faithfully represent a metric space as a subset of another space with controlled geometry, whose structure is well enough understood to successfully address the problem at hand. There is some obvious exibility in this approach: Both the choice of target space and the notion of faithfulness of an embedding can be adapted to the problem we wish to solve. Of course, once these choices are made, the main difficulty is the construction of the required embedding, and in the algorithmic context we have the additional burden of making sure that the embedding can be computed efficiently.The present paper is inspired by problems from mathematical analysis, but its main motivation is algorithmic. We introduce a new notion of embedding, called maximum gradient embeddings, which is just right for approximating a wide range of clustering problems. We will then provide optimal maximum gradient embeddings of general nite metric spaces, and use them to design the best known approximation algorithms for several natural clustering problems. We do not attempt to explore here all the possible applications of maximum gradient embeddings, and we suspect that there are many more situations in which our method is applicable. Indeed, rather than being encyclopedic, the main emphasis of the present paper is that these embeddings yield a generic approach to many problems, and we give some example that illustrate this fact. In addition, our work raises interesting algorithmic questions which deserve further investigation.The exibility that was described above in the choice of notions of embedding has been exploited to great success by numerous authors in the past four decades. Due to the vast amount of work on this topic, we will *

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“…It is known [10,15] that the properties of such partitions near x depend on the local value log |B(x,2τ )| |B(x,τ )| . Following [22], this relationship is used in [24] to prove (roughly) that, given the maps {ϕ τ } τ ≥0 defined above, there exists a map ϕ :…”

Section: Algorithmic Applicationmentioning

confidence: 99%

“…It is a concatenation of the result of Arora, Rao, and Vazirani [5], its strengthening by Lee [24], and the "reweighting" method of Chawla, Gupta, and Räcke [11], who use it in conjunction with [22] to achieve distortion O (log n) 3 4 . For the sake of completeness, we present below a sketch of the proof of Theorem 3.1.…”

Section: Single Scale Embeddingsmentioning

confidence: 99%

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2008

J. Amer. Math. Soc.

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Measured Descent: A New Embedding Method for Finite Metrics

Cited by 86 publications

References 29 publications

Online Embeddings

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Maximum gradient embeddings and monotone clustering

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