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Dimensionality Reductions That Preserve Volumes and Distance to Affine Spaces, and Their Algorithmic Applications

  • Avner Magen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2483)

Abstract

Let X be a subset of n points of the Euclidean space, and let 0 < ɛ < 1. A classical result of Johnson and Lindenstrauss [JL84] states that there is a projection of X onto a subspace of dimension O(ɛ-2 log n), with distortion ≤ 1 + ɛ. Here we show a natural extension of the above result, to a stronger preservation of the geometry of finite spaces. By a k-fold increase of the number of dimensions used compared to [JL84], a good preservation of volumes and of distances between points and affine spaces is achieved. Specifically, we show it is possible to embed a subset of size n of the Euclidean space into a O-2 klogn)-dimensional Euclidean space, so that no set of size sk changes its volume by more than (1+ɛ)s-1. Moreover, distances of points from affine hulls of sets of at most k-1 points in the space do not change by more than a factor of 1 + ɛ. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number of dimensions as the one used in [JL84]. Our method can be applied to many problems with high-dimensional nature such as Projective Clustering and Approximated Nearest Affine Neighbor Search. In particular, it shows a first poly-logarithmic query time approximation algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced by Feige [Fei00], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph.

Keywords

Euclidean Space Dimensionality Reduction Query Point Random Projection Dimensional Euclidean Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Avner Magen
    • 1
  1. 1.NEC Research InstitutePrinceton

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