J. Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93:333--344, 1996. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
On the Impossibility of Dimension Reduction in l_1 - Brinkman, Charikar (2003) (1 citation) (Correct)
....[10] gives a very good overview of the results in this area, particularly for isometric embeddings into # 1 and # 2 . It is known that dimension reduction is not possible in the # norm. In general, we need# n) dimensions to represent a set of n points in # with any distortion less than 3 [4, 21]. The only known dimensionality reduction theorem for # 1 is due to Indyk [14] He showed that there is an embedding from # 1 to # 1 with d # = log 1 #) O(1 #) such that distances do not increase with probability # and distances do not decrease by a factor (1 #) with probability 1 #. ....
J. Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93:333--344, 1996.
Embedding tree metrics into low dimensional Euclidean spaces - Gupta (2000) (9 citations) (Correct)
....space R d equipped with one of the p norms, kxk p = P i x p i ) 1=p . We shall refer to such spaces as d p . Thus, for example, d 2 is just standard d dimensional Euclidean space. Elegant techniques are known for embedding finite metrics into these spaces with low distortion [2, 8, 11]. Unfortunately, these techniques typically do not allow the dimension d of the target space to be restricted. On the other hand, many of the algorithms which use these embedding techniques (e.g. those for clustering [8] and finding approximate nearest neighbors [6] have performance guarantees ....
....space with distortion O(n 2=d (log n) 3=2 = p d) by first embedding it into a large number of dimensions and then projecting down onto a random d dimensional subspace. He also showed that this result is almost optimal for arbitrary metrics. Other aspects of this tradeoff have been studied in [1, 11]. Unfortunately, the analyses for these embedding techniques give a uniform bound on the distortion introduced. Hence this bound is as bad as the distortion incurred by the worst case metric, and gives no indication of whether special classes of metrics can be em bedded with much lesser ....
Jir'i Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel J. of Math., 93:333--344, 1996.
On Approximate Nearest Neighbors in Non-Euclidean Spaces - Indyk (1998) (10 citations) (Correct)
....the above result. One of them is that any finite metric space can be embedded in l d 1 with finite d and no distortion. Moreover, if we allow a small multiplicative distortion t, then any n point metric space can be embedded in l d 1 with small dimension d = O(n 1 b(t 1) 2c ) [Mat96]. By exploiting this property we obtain a tc(d; ae) approximation algorithm for a product of k arbitrary finite metrics of size s with query time O(ks 1 b(t 1) 2c ) and storage O(ks 1 b(t 1) 2c n 1 ae ) with polynomial preprocessing (a product of given metric spaces M 1 ; ....
....norms. The algorithm of [Kl97] is based random projections, which are not well defined for l p norms with p 2. The first algorithm of [IM98] uses the Johnson Lindenstrauss Lemma [JL84] to reduce the dimensionality to O(log n) this lemma provably does not hold for other norms (like l 1 [Mat96, Mat]) The algorithm of [KOR98] and the second result of [IM98] solve the problem by embedding the original norm into the Hamming space; again it is known that such an embedding is not possible for the l 1 norm. 2 Preliminaries In this section we introduce the notions and notation used later in the ....
J. Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93:333--344, 1996.
Approximate Distance Oracles - Thorup, Zwick (2001) (33 citations) (Correct)
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J. Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93:333--344, 1996.
Dimension Reduction in the l1 norm - Charikar, Sahai (2002) (6 citations) (Correct)
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J. Matousek. On the distortion required for embedding finite metric spaces into normed spaces. Israel Journal of Mathematics, 93:333--344, 1996.
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