### Abstract

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(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.

Original language | English (US) |
---|---|

Pages (from-to) | 1-36 |

Number of pages | 36 |

Journal | Foundations of Computational Mathematics |

Volume | 13 |

Issue number | 1 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

- Alternative duals
- Compressed sensing
- Finite frames
- Quantization
- Random frames

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Computational Mathematics
- Computational Theory and Mathematics

### Cite this

*Foundations of Computational Mathematics*,

*13*(1), 1-36. https://doi.org/10.1007/s10208-012-9140-x

**Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements.** / Gunturk, C. Sinan; Lammers, M.; Powell, A. M.; Saab, R.; Yilmaz, Ö.

Research output: Contribution to journal › Article

*Foundations of Computational Mathematics*, vol. 13, no. 1, pp. 1-36. https://doi.org/10.1007/s10208-012-9140-x

}

TY - JOUR

T1 - Sobolev Duals for Random Frames and ΣΔ Quantization of Compressed Sensing Measurements

AU - Gunturk, C. Sinan

AU - Lammers, M.

AU - Powell, A. M.

AU - Saab, R.

AU - Yilmaz, Ö

PY - 2013

Y1 - 2013

N2 - 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(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.

AB - 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(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.

KW - Alternative duals

KW - Compressed sensing

KW - Finite frames

KW - Quantization

KW - Random frames

UR - http://www.scopus.com/inward/record.url?scp=84872557345&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84872557345&partnerID=8YFLogxK

U2 - 10.1007/s10208-012-9140-x

DO - 10.1007/s10208-012-9140-x

M3 - Article

AN - SCOPUS:84872557345

VL - 13

SP - 1

EP - 36

JO - Foundations of Computational Mathematics

JF - Foundations of Computational Mathematics

SN - 1615-3375

IS - 1

ER -