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ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk

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Automata, Languages, and Programming (ICALP 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7965))

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Abstract

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 ∈ ℝN, 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.

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Author information

Authors and Affiliations

  1. University of Michigan, USA

    Anna C. Gilbert & Martin J. Strauss

  2. University at Buffalo (SUNY), USA

    Hung Q. Ngo & Atri Rudra

  3. Bar-Ilan University, Israel

    Ely Porat

Authors
  1. Anna C. Gilbert
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  2. Hung Q. Ngo
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  3. Ely Porat
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  4. Atri Rudra
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  5. Martin J. Strauss
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Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Postboks 7803, 5020, Bergen, Norway

    Fedor V. Fomin

  2. Faculty of Computing, University of Latvia, Raina bulv. 19, 1586, Riga, Latvia

    Rūsiņš Freivalds

  3. Department of Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK

    Marta Kwiatkowska

  4. Faculty of Mathematics and Computer Science, Weizmann Institute of Science, POB 26, 76100, Rehovot, Israel

    David Peleg

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Gilbert, A.C., Ngo, H.Q., Porat, E., Rudra, A., Strauss, M.J. (2013). ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds) Automata, Languages, and Programming. ICALP 2013. Lecture Notes in Computer Science, vol 7965. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39206-1_39

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  • DOI: https://doi.org/10.1007/978-3-642-39206-1_39

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-39205-4

  • Online ISBN: 978-3-642-39206-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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