Embedding Finite Metric Spaces in Low Dimension (original) (raw)

Advances in Mathematics, 2011

Embedding metric spaces in their intrinsic dimension

… of the nineteenth annual ACM-SIAM …, 2008

A fundamental question of metric embedding is whether the metric dimension of a metric space is related to its intrinsic dimension. That is whether the dimension in which it can be embedded in some real normed space is implied by the intrinsic dimension which is reflected by the inherent geometry of the space. The existence of such an embedding was conjectured by Assouad and was later posed as an open problem by others. This question is tightly related to a major goal of many practical application fields: developing tools to represent intrinsically low dimensional metric data sets in a succinct manner. In this paper we give the first algorithmic technique with formal guarantees for finding faithful and low dimensional representations of data lying in high dimensional space. Our main theorem states that every finite metric space X embeds into Euclidean space with dimension O(dim(X)/ ) and distortion O(log 1+ε n), where dim(X) is the doubling dimension of the space X. Moreover, we show that X can be embedded into dimensionÕ(dim(X)) with constant average distortion and q -distortion for any q < ∞. Our technique also provides a dimension-distortion tradeoff and an extension of Assouad's theorem, providing distance oracles that improve known construction when dim(X) = o(log |X|).

On embeddings of finite metric spaces

2015 IEEE 13th International Scientific Conference on Informatics, 2015

Let ⟨X, ϱ⟩ be a finite metric space, and for a natural number d, let R d be the real d-dimensional vector space endowed with its usual Euclidean metric. We interested in estimations for d such that ⟨X, ϱ⟩ can be "embedded" in some sense into R d. This classical topic of functional analysis recently has received renewed impetus motivated by several problems of theoretical computer science. We will recall some of these problems which also help us to find the "good" notion of embeddings and announce some recently obtained related results.

Inapproximability for metric embeddings into mathbbRd\mathbb{R}^{d}mathbbRd

Transactions of the American Mathematical Society, 2010

We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into R d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1/12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1/(22d−10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Väisälä). For d ≥ 3, we obtain a stronger inapproximability result by a different reduction: assuming P =NP, no polynomial-time algorithm can distinguish between spaces embeddable in R d with constant distortion from spaces requiring distortion at least n c/d , for a constant c > 0. The exponent c/d has the correct order of magnitude, since every n-point metric space can be embedded in R d with distortion O(n 2/d log 3/2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in R 2 , while all not too large subspaces of it embed almost isometrically.

On Embedding of Finite Metric Spaces into Hilbert Space

2005

Metric embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The main criteria for the quality of an embedding is its average distortion over all pairs. A celebrated theorem of Bourgain states that every finite metric space on n points embeds in Euclidean space with O(logn) distortion. Bourgain's result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, is this the case for the average distortion? Our main result is a strengthening of Bourgain's theorem providing an embedding with constant average distortion for arbitrary metric spaces. In fact, our embedding possesses a much stronger property. We define the 'q-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible 'q-distortion for all 1 • q • 1 simultaneously.

Advances in Metric Embedding Theory Extended

2006

Metric Embedding plays an important role in a vast range of application areas such as computer vision, computational biology, machine learning, networking, statistics, and mathematical psychology, to name a few. The theory of metric embedding received much attention in recent years by mathematicians as well as computer scientists and has been applied in many algorithmic applications. A cornerstone of the field is a celebrated theorem of Bourgain which states that every finite metric space on n points embeds in Euclidean space with O(logn) distortion. Bourgain’s result is best possible when considering the worst case distortion over all pairs of points in the metric space. Yet, it is possible that an embedding can do much better in terms of the average distortion. Indeed, in most practical applications of metric embedding the main criteria for the quality of an embedding is its average distortion over all pairs. In this paper we provide an embedding with constant average distortion f...

Metric Embeddings with Relaxed Guarantees

2005

We consider the problem of embedding finite metrics with slack: we seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed.

Approximation algorithms for low-distortion embeddings into low-dimensional spaces

2005

We present several approximation algorithms for the problem of embedding metric spaces into a line, and into the two-dimensional plane. We give an O( √ n)approximation algorithm for the problem of finding a line embedding of a metric induced by a given unweighted graph, that minimizes the (standard) multiplicative distortion. For the same problem, we give an exact algorithm, with running-time exponential in the distortion. We complement these results by showing that the problem is NP-hard to α-approximate, for some constant α > 1.

Optimal Embeddings of Finite Metric Spaces Into Graphs

Anadolu Üniversitesi Bilim Ve Teknoloji Dergisi - B Teorik Bilimler, 2015

We consider the embedding of a finite metric space into a weighted graph in such a way that the total weight of the edges is minimal. We discuss metric spaces with = 3,4,5 points in detail and show that the already known classification for these cases can be obtained by simple operations on the associated graph of the given metric space.

Embedding Finite Metric Spaces in Low Dimension (original) (raw)

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