Rademacher-Sketch: A Dimensionality-Reducing Embedding for Sum-Product Norms, with an Application to Earth-Mover Distance

  • Elad Verbin
  • Qin Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


Consider a sum-product normed space, i.e. a space of the form \(Y=\ell_1^n \otimes X\), where X is another normed space. Each element in Y consists of a length-n vector of elements in X, and the norm of an element in Y is the sum of the norms of its coordinates. In this paper we show a constant-distortion embedding from the normed space \(\ell_1^n \otimes X\) into a lower-dimensional normed space \(\ell_1^{n'} \otimes X\), where n′ ≪ n is some value that depends on the properties of the normed space X (namely, on its Rademacher dimension). In particular, composing this embedding with another well-known embedding of Indyk [18], we get an O(1/ε)-distortion embedding from the earth-mover metric EMD Δ on the grid [Δ]2 to \(\ell_1^{\Delta^{O(\epsilon )}} \otimes {\sf{EEMD}}_{\Delta^{\epsilon }}\) (where EEMD is a norm that generalizes earth-mover distance). This embedding is stronger (and simpler) than the sketching algorithm of Andoni et al [4], which maps EMD Δ with O(1/ε) approximation into sketches of size Δ O(ε).


Normed Space Hash Function Earth Mover Sample Level Pseudorandom Generator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Elad Verbin
    • 1
  • Qin Zhang
    • 2
  1. 1.MADALGO and CTICAarhus UniversityDenmark
  2. 2.MADALGOAarhus UniversityDenmark

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