Advances in metric embedding theory

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Advances in metric embedding theory (2006)

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by Ittai Abraham , Yair Bartal , Ofer Neiman

Venue:	IN STOC ’06: PROCEEDINGS OF THE THIRTY-EIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
Citations:	36 - 13 self

BibTeX

@INPROCEEDINGS{Abraham06advancesin,
    author = {Ittai Abraham and Yair Bartal and Ofer Neiman},
    title = {Advances in metric embedding theory},
    booktitle = {IN STOC ’06: PROCEEDINGS OF THE THIRTY-EIGHTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING},
    year = {2006},
    pages = {271--286},
    publisher = {ACM Press}
}

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Abstract

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(log n) 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 for arbitrary metric spaces, while maintaining the same worst case bound provided by Bourgain’s theorem. In fact, our embedding possesses a much stronger property. We define the ℓq-distortion of a uniformly distributed pair of points. Our embedding achieves the best possible ℓq-distortion for all 1 ≤ q ≤ ∞ simultaneously. These results have several algorithmic implications, e.g. an O(1) approximation for the unweighted uncapacitated quadratic assignment problem. The results are based on novel embedding methods which improve on previous methods in another important aspect: the dimension. The dimension of an embedding is of very high importance in particular in applications and much effort has been invested in analyzing it. However, no previous result im-

Keyphrases

average distortion important role bourgain theorem case bound celebrated theorem point embeds metric embedding finite metric space constant average distortion possible q-distortion machine learning arbitrary metric space computational biology much effort computer vision many algorithmic application metric space previous result im much attention main criterion computer scientist mathematical psychology practical application vast range case distortion important aspect high importance previous method application area recent year several algorithmic implication bourgain result unweighted uncapacitated quadratic assignment problem euclidean space

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