Date: 27 Oct 2012
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.
Communicated by Emmanuel Candès.
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- Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements
Foundations of Computational Mathematics
Volume 13, Issue 1 , pp 1-36
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Finite frames
- Random frames
- Alternative duals
- Compressed sensing
- Industry Sectors
- Author Affiliations
- 1. Courant Institute of Mathematical Sciences, New York University, New York, USA
- 2. University of North Carolina, Wilmington, USA
- 3. Vanderbilt University, Nashville, USA
- 4. Duke University, Durham, USA
- 5. University of British Columbia, Vancouver, Canada