Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements
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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 ℓ 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.
KeywordsQuantization Finite frames Random frames Alternative duals Compressed sensing
Mathematics Subject Classification41A46 94A12
The authors would like to thank Ronald DeVore and Vivek Goyal for valuable discussions. We thank the American Institute of Mathematics and Banff International Research Station for hosting two meetings where this work was initiated. This work was supported in part by: National Science Foundation Grant CCF-0515187 (Güntürk), Alfred P. Sloan Research Fellowship (Güntürk), National Science Foundation Grant DMS-0811086 (Powell), the Academia Sinica Institute of Mathematics in Taipei, Taiwan (Powell), a Pacific Century Graduate Scholarship from the Province of British Columbia through the Ministry of Advanced Education (Saab), a UGF award from the UBC (Saab), an NSERC (Canada) Discovery Grant (Yılmaz), and an NSERC Discovery Accelerator Award (Yılmaz).
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