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Dimensionality Reduction: beyond the JohnsonLindenstrauss bound
"... Dimension reduction of metric data has become a useful technique with numerous applications. The celebrated JohnsonLindenstrauss lemma states that any npoint subset of Euclidean space can be embedded in O(ɛ −2 log n) dimension with 1 + ɛ distortion. This bound is known to be nearly tight. In many ..."
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Dimension reduction of metric data has become a useful technique with numerous applications. The celebrated JohnsonLindenstrauss lemma states that any npoint subset of Euclidean space can be embedded in O(ɛ −2 log n) dimension with 1 + ɛ distortion. This bound is known to be nearly tight. In many applications the demand that all distances would be nearly observed is too strong. In this paper we show that indeed under natural relaxations of the goal of the embedding, an improved dimension reduction is possible where the target dimension is independent of n. Our main result can be viewed as a local dimension reduction. There are a variety of empirical situations in which small distances are meaningful and reliable, but larger ones are not. Such situations arise in source coding, image processing, computational biology, and other applications, and are the motivation for widelyused heuristics such as Isomap and Locally Linear Embedding. Pursuing a line of work begun by Whitney, Nash showed that every C 1 manifold of dimension d can be embedded in R 2d+2 in such a manner that the local structure at each point is preserved isometrically. Our work is an analog of Nash’s for discrete subsets of Euclidean space. For perfect preservation of infinitesimal neighborhoods we substitute nearisometric embedding of neighborhoods of bounded
On Low Dimensional Local Embeddings
, 2010
"... We study the problem of embedding metric spaces into low dimensional Lp spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into L O(ep log 2 k) p with klocal distortion of O((log k)/p). We also O(log k)/ɛ3 show that ..."
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We study the problem of embedding metric spaces into low dimensional Lp spaces while faithfully preserving distances from each point to its k nearest neighbors. We show that any metric space can be embedded into L O(ep log 2 k) p with klocal distortion of O((log k)/p). We also O(log k)/ɛ3 show that any ultrametric can be embedded into Lp with klocal distortion 1 + ɛ. Our embedding results have immediate applications to local Distance Oracles. We show how to preprocess a graph in polynomial time to obtain a data structure of O(nk1/t log 2 k) bits, such that distance queries from any node to its k nearest neighbors can be answered with stretch O(t).
UltraLowDimensional Embeddings for Doubling Metrics
"... We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argumen ..."
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We consider the problem of embedding a metric into lowdimensional Euclidean space. The classical theorems of Bourgain and of Johnson and Lindenstrauss imply that any metric on n points embeds into an O(log n)dimensional Euclidean space with O(log n) distortion. Moreover, a simple “volume ” argument shows that this bound is nearly tight: the uniform metric on n points requires Ω(log n / log log n) dimensions to embed with logarithmic distortion. It is natural to ask whether such a volume restriction is the only hurdle to lowdimensional lowdistortion embeddings. Do doubling metrics, which do not have large uniform submetrics, embed in low dimensional Euclidean spaces with small distortion? In this paper, we answer the question positively and show that any doubling metric embeds into O(log log n) dimensions with o(log n) distortion. In fact, we give a suite of embeddings with a smooth tradeoff between distortion and dimension: given an npoint metric (V, d) with doubling dimension dimD, and any target dimension T in the range Ω(dimD log log n) ≤ T ≤ O(log n), we embed the metric into Euclidean space R T with O(log n � dimD /T) distortion.
Fast ckr partitions of sparse graphs
 Chicago Journal of Theoretical Computer Science
"... We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1 ..."
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We present fast algorithms for constructing probabilistic embeddings and approximate distance oracles in sparse graphs. The main ingredient is a fast algorithm for sampling the probabilistic partitions of Calinescu, Karloff, and Rabani in sparse graphs. 1
The Art of Metric Embeddings — A technique oriented approach
, 2007
"... During the last two decades, embeddings into finite metric spaces has emerged as an important are of research in both discrete mathematics and computer science. The powerful techniques underlying this research find numerous applications in the design and analysis of algorithms. In this survey, we tr ..."
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During the last two decades, embeddings into finite metric spaces has emerged as an important are of research in both discrete mathematics and computer science. The powerful techniques underlying this research find numerous applications in the design and analysis of algorithms. In this survey, we try to cover the basic approach and techniques, as well as some of their applications. Unlike most surveys, we do not put as much emphasis on the results achieved in this burgeoning field. The reason is that metric embedding is such a wide domain that it seems impossible for us to present a comprehensive review in a short survey. Instead, we try to uncover the most essential ideas and techniques used in the field and relate them to similar methods used in other areas. We hope this approach would be useful to people who wish to understand why and how metric embeddings can yield so many deep and surprising results in algorithm design.
Algorithms and Models for Problems in Networking
, 2010
"... Many interesting theoretical problems arise from computer networks. In this thesis we will consider three of them: algorithms and data structures for problems involving distances in networks (in particular compact routing schemes, distance labels, and distance oracles), algorithms for wireless capac ..."
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Many interesting theoretical problems arise from computer networks. In this thesis we will consider three of them: algorithms and data structures for problems involving distances in networks (in particular compact routing schemes, distance labels, and distance oracles), algorithms for wireless capacity and scheduling problems, and algorithms for optimizing iBGP overlays in autonomous systems on the Internet. While at first glance these problems may seem extremely different, they are similar in that they all attempt to look at a previously studied networking problem in new, more realistic frameworks. In other words, they are all as much about new models for old problems as they are about new algorithms. In this thesis we will define these models, design algorithms for them, and prove hardness and impossibility results for these three types of problems. viAcknowledgments This thesis would have been impossible without the guidance of my advisor, Anupam Gupta. While we may not have written many papers together, he has been an invaluable mentor who always has good ideas and interesting thoughts. I was
Graph Augmentation via Metric Embedding
"... Abstract. Kleinberg [17] proposed in 2000 the first random graph model achieving to reproduce small world navigability, i.e. the ability to greedily discover polylogarithmic routes between any pair of nodes in a graph, with only a partial knowledge of distances. Following this seminal work, a major ..."
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Abstract. Kleinberg [17] proposed in 2000 the first random graph model achieving to reproduce small world navigability, i.e. the ability to greedily discover polylogarithmic routes between any pair of nodes in a graph, with only a partial knowledge of distances. Following this seminal work, a major challenge was to extend this model to larger classes of graphs than regular meshes, introducing the concept of augmented graphs navigability. In this paper, we propose an original method of augmentation, based on metrics embeddings. Precisely, we prove that, for any ε>0, any graph G such that its shortest paths metric admits an embedding of distorsion γ into R d can be augmented by one link per node such that greedy routing computes paths of expected length O ( 1 ε γd log 2+ε n) between any pair of nodes with the only knowledge of G. Ourmethod isolates all the structural constraints in the existence of a good quality embedding and therefore enables to enlarge the characterization of augmentable graphs.