منابع مشابه
Metric Transforms and Euclidean Embeddings
It is proved that if 0 < c < 0.72/« then for any «-point metric space (X, d), the metric space (X,dc) is isometrically embeddable into a Euclidean space. For 6-point metric space, c = j log2 | is the largest exponent that guarantees the existence of isometric embeddings into a Euclidean space. Such largest exponent is also determined for all «-point graphs with "truncated distance".
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When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small (“snowflake”) deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, lik...
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It was recently observed by de Vienne et al. (Syst Biol 60(6):826-832, 2011) that a simple square root transformation of distances between taxa on a phylogenetic tree allowed for an embedding of the taxa into Euclidean space. While the justification for this was based on a diffusion model of continuous character evolution along the tree, here we give a direct and elementary explanation for it t...
متن کاملEuclidean Embeddings that Preserve Volumes
Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by a factor of 1 + ε at most. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs, but all subsets of at most k p...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1990
ISSN: 0002-9947
DOI: 10.2307/2001482