Journal
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
Volume 13, Issue 1, Pages 1-36Publisher
SPRINGER
DOI: 10.1007/s10208-012-9140-x
Keywords
Quantization; Finite frames; Random frames; Alternative duals; Compressed sensing
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Quantization of compressed sensing measurements is typically justified by the robust recovery results of CandSs, 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=I broken vertical bar x of a k-sparse signal xaae (N) , where I broken vertical bar satisfies the restricted isometry property, then the approximate recovery x (#) via a (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 rth-order I I pound (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 delta(k/m)((r-1/2)alpha) for any 0 < 1, if ma parts per thousand(3) (r,alpha) k(logN)(1/(1-alpha)). The result holds with high probability on the initial draw of the measurement matrix I broken vertical bar from the Gaussian distribution, and uniformly for all k-sparse signals x whose magnitudes are suitably bounded away from zero on their support.
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