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Ordinal embedding: Approximation algorithms and dimensionality reduction

Bǎdoiu, M ; Sharif University of Technology | 2008

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  1. Type of Document: Article
  2. DOI: 10.1007/978-3-540-85363-3_3
  3. Publisher: 2008
  4. Abstract:
  5. This paper studies how to optimally embed a general metric, represented by a graph, into a target space while preserving the relative magnitudes of most distances. More precisely, in an ordinal embedding, we must preserve the relative order between pairs of distances (which pairs are larger or smaller), and not necessarily the values of the distances themselves. The relaxation of an ordinal embedding is the maximum ratio between two distances whose relative order is inverted by the embedding. We develop polynomial-time constant-factor approximation algorithms for minimizing the relaxation in an embedding of an unweighted graph into a line metric and into a tree metric. These two basic target metrics are particularly important for representing a graph by a structure that is easy to understand, with applications to visualization, compression, clustering, and nearest-neighbor searching. Along the way, we improve the best known approximation factor for ordinally embedding unweighted trees into the line down to 2. Our results illustrate an important contrast to optimal-distortion metric embeddings, where the best approximation factor for unweighted graphs into the line is O(n 1/2), and even for unweighted trees into the line the best is . We also show that Johnson-Lindenstrauss-type dimensionality reduction is possible with ordinal relaxation and l1 metrics (and lp metrics with 1 ≤ p ≤ 2), unlike metric embedding of l1 metrics. © 2008 Springer-Verlag Berlin Heidelberg
  6. Keywords:
  7. Approximation factors ; Best approximations ; Combinatorial op-timization problem ; Dimensionality reduction ; Distortion metric ; Maximum ratio ; Nearest neighbors ; Polynomial-time ; Relative magnitudes ; Relative order ; Target space ; Unweighted graphs ; Canning ; Combinatorial mathematics ; Combinatorial optimization ; Graph theory ; Learning systems ; Optimization ; Polynomial approximation ; Trees (mathematics) ; Approximation algorithms
  8. Source: 11th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2008 and 12th International Workshop on Randomization and Computation, RANDOM 2008, Boston, MA, 25 August 2008 through 27 August 2008 ; Volume 5171 LNCS , 2008 , Pages 21-34 ; 03029743 (ISSN) ; 9783540853626 (ISBN)
  9. URL: https://link.springer.com/chapter/10.1007/978-3-540-85363-3_3