Abstract
Quantization of compressed sensing measurements is typically justified by the robust recovery results of Candes, Romberg and Tao, and of Donoho. These results guarantee that if a uniform quantizer of step size ź is used to quantize m measurements y=źx of a k-sparse signal xźźN, where ź satisfies the restricted isometry property, then the approximate recovery x# via l1-minimization is within O(ź) 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 ź(k/m)(rź1/2)ź for any 0<ź<1, if mźr,źk(logN)1/(1źź). The result holds with high probability on the initial draw of the measurement matrix ź 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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