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Combinatorica

, Volume 30, Issue 1, pp 47–68 | Cite as

Almost Euclidean subspaces of ℓ 1 N VIA expander codes

  • Venkatesan Guruswami
  • James R. Lee
  • Alexander Razborov
Article

Abstract

We give an explicit (in particular, deterministic polynomial time) construction of subspaces X⊆ℝ N of dimension (1−o(1))N such that for every xX,
$$ (\log N)^{ - O(\log \log \log N)} \sqrt N \left\| x \right\|_2 \leqslant \left\| x \right\|_1 \leqslant \sqrt N \left\| x \right\|_2 $$
. If we are allowed to use N 1/log logN N o(1) random bits and dim(X) ⩾ (1−η)N for any fixed constant η, the lower bound can be further improved to \( (\log N)^{ - O(1)} \sqrt N \left\| x \right\|_2 \).

Through known connections between such Euclidean sections of ℓ1 and compressed sensing matrices, our result also gives explicit compressed sensing matrices for low compression factors for which basis pursuit is guaranteed to recover sparse signals. Our construction makes use of unbalanced bipartite graphs to impose local linear constraints on vectors in the subspace, and our analysis relies on expansion properties of the graph. This is inspired by similar constructions of error-correcting codes.

Mathematics Subject Classification (2000)

68R05 68P30 51N20 

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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2010

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • James R. Lee
    • 2
    • 3
  • Alexander Razborov
    • 4
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Computer ScienceUniversity of ChicagoChicagoUSA
  3. 3.Steklov Mathematical InstituteMoscowRussia
  4. 4.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA

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