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Sobolev Duals for Random Frames and Sigma-Delta Quantization of Compressed Sensing Measurements

C. S. Güntürk,Alexander M. Powell,Rayan Saab,Özgür Yilmaz

Published 2010 in arXiv.org

ABSTRACT

Quantization of compressed sensing measurements is typically justified by the robust recovery results of Cand\`es, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size $\delta$ is used to quantize $m$ measurements $y = \Phi x$ of a $k$-sparse signal $x \in \R^N$, where $\Phi$ satisfies the restricted isometry property, then the approximate recovery $x^#$ via $\ell_1$-minimization is within $O(\delta)$ of $x$. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an $r$th order $\Sigma\Delta$ quantization scheme with the same output alphabet is used to quantize $y$, then there is an alternative recovery method via Sobolev dual frames which guarantees a reduction of the approximation error by a factor of $(m/k)^{(r-1/2)\alpha}$ for any $0 < \alpha < 1$, if $m \gtrsim_r k (\log N)^{1/(1-\alpha)}$. The result holds with high probability on the initial draw of the measurement matrix $\Phi$ from the Gaussian distribution, and uniformly for all $k$-sparse signals $x$ that satisfy a mild size condition on their supports.

PUBLICATION RECORD

  • Publication year

    2010

  • Venue

    arXiv.org

  • Publication date

    2010-02-01

  • Fields of study

    Mathematics, Computer Science

  • Identifiers
    arXiv 1002.0182
  • External record

    Open on Semantic Scholar

  • Source metadata

    Semantic Scholar

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