2/ℓ2-Foreach Sparse Recovery with Low Risk

  • Anna C. Gilbert
  • Hung Q. Ngo
  • Ely Porat
  • Atri Rudra
  • Martin J. Strauss
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)


In this paper, we consider the “foreach” sparse recovery problem with failure probability p. The goal of the problem is to design a distribution over m ×N matrices Φ and a decoding algorithm A such that for every x ∈ ℝ N , we have with probability at least 1 − p
$$\|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$
where x k is the best k-sparse approximation of x.

Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k) + log(1/p)) for \(2^{-\Theta(N)}\leqslant p <1\). Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p = Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries.


Failure Probability Recovery Algorithm Recovery Problem Sparse Recovery Heavy Hitter 
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 2013

Authors and Affiliations

  • Anna C. Gilbert
    • 1
  • Hung Q. Ngo
    • 2
  • Ely Porat
    • 3
  • Atri Rudra
    • 2
  • Martin J. Strauss
    • 1
  1. 1.University of MichiganUSA
  2. 2.University at Buffalo (SUNY)USA
  3. 3.Bar-Ilan UniversityIsrael

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