Foundations of Computational Mathematics

, Volume 13, Issue 1, pp 1-36

First online:

Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

  • C. S. GüntürkAffiliated withCourant Institute of Mathematical Sciences, New York University
  • , M. LammersAffiliated withUniversity of North Carolina
  • , A. M. PowellAffiliated withVanderbilt University
  • , R. SaabAffiliated withDuke University
  • , Ö. YılmazAffiliated withUniversity of British Columbia Email author 

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Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candès, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size δ is used to quantize m measurements y=Φx of a k-sparse signal x∈ℝ N , where Φ satisfies the restricted isometry property, then the approximate recovery x # via 1-minimization is within O(δ) of x. The simplest and commonly assumed approach is to quantize each measurement independently. In this paper, we show that if instead an rth-order ΣΔ (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 reduced approximation error that is of the order δ(k/m)(r−1/2)α for any 0<α<1, if m r,α k(logN)1/(1−α). The result holds with high probability on the initial draw of the measurement matrix Φ from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.


Quantization Finite frames Random frames Alternative duals Compressed sensing

Mathematics Subject Classification

41A46 94A12