Advances in metric embedding theory

Abraham, Ittai; Bartal, Yair; Neiman, Ofer

doi:10.1016/j.aim.2011.08.003

Cited by 82 publications

(221 citation statements)

References 44 publications

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“…Linial, London and Rabinovich also proved an Ω(log n) lower bound for the distortion. Further improvement was obtained by Abraham, Bartal and Neiman [6], who reduced the target dimension to O(log n), which is optimal, and also showed that the average (but not the worst case!) distortion can be made constant.…”

Section: mentioning

confidence: 99%

Optimizing Massive Distance Computations in Pattern Recognition

Faragó¹

2014

IJMLC

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Abstract-It is a common task in pattern recognition to evaluate the similarity of large data objects. These are often represented by high dimensional vectors. A frequently used mathematical model for evaluating their similarity is to view them as points (vectors) in a high dimensional space, and compute their distances from each other. The "distance," however, can be defined in a very complicated way, it may be much more complex than the well known Euclidean distance. Therefore, the algorithmic bottleneck often becomes the number of distance computations that need to be carried out. We consider the case when we have to compute all the distances between n objects, where n is large. Without any shortcuts it takes n(n − 1)/2 = O(n 2 ) distance computations. In those applications where the distances are complicated, being defined by sophisticated algorithms (such as in speech and image recognition), a quadratically growing number of distance computations becomes a severe bottleneck. We prove the following general result that can help eliminating the bottleneck: for a large and general class of distances it is possible to obtain a very close approximation of each of the O(n 2 ) pairwise distances of n objects by doing only a linear number distance computations, which is optimal with respect to the order of magnitude. Moreover, the approximation factor can be made arbitrarily close to 1, making the approximation error negligible. The needed side computations to achieve this reduction can also be done in polynomial time.

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Section: mentioning

confidence: 99%

Optimizing Massive Distance Computations in Pattern Recognition

Faragó¹

2014

IJMLC

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“…Yet, as already mentioned above, when communication is required between specific pairs of nodes, the cost of routing through the MST may be extremely high, even when their real distance is small. However, in practice it is the average distortion, rather than the worst-case distortion, that is often used as a practical measure of quality, as has been a major motivation behind the initial work of [21,3,4]. As noted above, the MST still fails even in this relaxed measure.…”

Section: Introductionmentioning

confidence: 99%

“…We also obtain a probabilistic embedding devising a distribution over (light) spanning trees with polylog(1/ )/ρ 2 scaling distortion, thus providing constant bounds on all fixed moments of the distortion (i.e., the l q -distortion [3] for fixed q).…”

Section: Introductionmentioning

confidence: 99%

“…Here we show a light spanner construction with prioritized distortion at most O(log 2 j)/ρ 2 . We then show a connection between the notions of prioritized distortion and scaling distortion 3 Originally coined gracefully degrading embedding.…”

Section: Introductionmentioning

confidence: 99%

“…This notion, first introduced in [21] 3 , requires that for every 0 < < 1, the distortion of all but an -fraction of the pairs is bounded by the appropriate function of . In [3] it was shown that one may obtain bounds on the average distortion, as well as on higher moments of the distortion function, from bounds on the scaling distortion. Our scaling distortion bound for the constructed spanning tree is 4Õ (1/ √ )/ρ 2 , which is nearly tight as a function of [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

Bartal

Filtser

Neiman

2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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Minimum Spanning Trees of weighted graphs are fundamental objects in numerous applications. In particular in distributed networks, the minimum spanning tree of the network is often used to route messages between network nodes. Unfortunately, while being most efficient in the total cost of connecting all nodes, minimum spanning trees fail miserably in the desired property of approximately preserving distances between pairs. While known lower bounds exclude the possibility of the worst case distortion of a tree being small, it was shown in [4] that there exists a spanning tree with constant average distortion. Yet, the weight of such a tree may be significantly larger than that of the MST. In this paper, we show that any weighted undirected graph admits a spanning tree whose weight is at most (1 + ρ) times that of the MST, providing constant average distortion O(1/ρ 2 ). 1 The constant average distortion bound is implied by a stronger property of scaling distortion, i.e., improved distortion for smaller fractions of the pairs. The result is achieved by first showing the existence of a low weight spanner with small prioritized distortion, a property allowing to prioritize the nodes whose associated distortions will be improved. We show that prioritized distortion is essentially equivalent to coarse scaling distortion via a general transformation, which has further implications and may be of independent interest. In particular, we obtain an embedding for arbitrary metrics into Eu-

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A Fast Fiber k-Nearest-Neighbor Algorithm with Application to Group-Wise White Matter Topography Analysis

Wang¹,

Shi²

2019

Lecture Notes in Computer Science

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Advances in metric embedding theory

Cited by 82 publications

References 44 publications

Optimizing Massive Distance Computations in Pattern Recognition

Optimizing Massive Distance Computations in Pattern Recognition

On Notions of Distortion and an Almost Minimum Spanning Tree with Constant Average Distortion

A Fast Fiber k-Nearest-Neighbor Algorithm with Application to Group-Wise White Matter Topography Analysis

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