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

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 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(log N)^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.