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Embedding, Distance Estimation and Object Location in Networks
, 2006
"... Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into l ..."
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Concurrent with numerous theoretical results on metric embeddings, a growing body of research in the networking community has studied the distance matrix defined by nodetonode latencies in the Internet, resulting in a number of recent approaches that approximately embed this distance matrix into lowdimensional Euclidean space. A fundamental distinction between the theoretical approaches to embeddings and this recent Internetrelated work is that the latter operates under the additional constraint that it is only feasible to measure a linear number of node pairs, and typically in a highly structured way. Indeed, the most common framework here is a beaconbased approach: one randomly chooses a small number of nodes (’beacons’) in the network, and each node measures its distance to these beacons only. Moreover, beaconbased algorithms are also designed for the more basic problem of triangulation, in which one uses the triangle inequality to infer the distances that have not been measured. We give algorithms with provable performance guarantees for triangulation and embedding. We show that in addition to multiplicative error in the distances, performance guarantees for beaconbased algorithms typically must include a notion of ”slack ” – a certain fraction of all distances may be arbitrarily distorted. For arbitrary metrics, we give a beaconbased embedding algorithm that achieves constant distortion on a (1 − ɛ)fraction of distances; this provides some theoretical justification for the success of the recent
Sparsity and nonEuclidean embeddings
, 2012
"... We present a relation between sparsity and nonEuclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of `np, p> 0, into various type of Banach or quasiBanach spaces. In particular, for 0 < r < p < 2 with ..."
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We present a relation between sparsity and nonEuclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables to construct embeddings of `np, p> 0, into various type of Banach or quasiBanach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed `np into ` (1+η)n r, with sharp polynomial bounds in η> 0. 1
Random embedding of l np into l Nr
"... For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator S: `np → RN such that for every 0 < r < p < 2 with r ≤ 1, the operator Sr = S: `np → `Nr satisfies with overwhelming probability that ‖Sr ‖ ‖(Sr)−1ImS ‖ ≤ C(p, r)n/(N−n), where C( ..."
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For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator S: `np → RN such that for every 0 < r < p < 2 with r ≤ 1, the operator Sr = S: `np → `Nr satisfies with overwhelming probability that ‖Sr ‖ ‖(Sr)−1ImS ‖ ≤ C(p, r)n/(N−n), where C(p, r)> 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.