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

• Published in: Foundations of computational mathematics Vol. 13; no. 1; pp. 1 - 36
• Main Authors: Güntürk, C. S., Lammers, M., Powell, A. M., Saab, R., Yılmaz, Ö.
• Format: Journal Article
• Language: English
• Published: New York Springer-Verlag 01.02.2013; Springer
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Economics; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Signal processing; United States; Finite frames; Random frames; 41A46; Quantization; 94A12; Alternative duals; Compressed sensing
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-012-9140-x

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 r th-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 (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.