Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements
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- Güntürk, C.S., Lammers, M., Powell, A.M. et al. Found Comput Math (2013) 13: 1. doi:10.1007/s10208-012-9140-x
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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∈ℝN, where Φ satisfies the restricted isometry property, then the approximate recovery x# via ℓ1-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,αk(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.