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

- 1k Downloads
- 23 Citations

## 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 *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.

## Keywords

Quantization Finite frames Random frames Alternative duals Compressed sensing## Mathematics Subject Classification

41A46 94A12## Notes

### Acknowledgements

The authors would like to thank Ronald DeVore and Vivek Goyal for valuable discussions. We thank the American Institute of Mathematics and Banff International Research Station for hosting two meetings where this work was initiated. This work was supported in part by: National Science Foundation Grant CCF-0515187 (Güntürk), Alfred P. Sloan Research Fellowship (Güntürk), National Science Foundation Grant DMS-0811086 (Powell), the Academia Sinica Institute of Mathematics in Taipei, Taiwan (Powell), a Pacific Century Graduate Scholarship from the Province of British Columbia through the Ministry of Advanced Education (Saab), a UGF award from the UBC (Saab), an NSERC (Canada) Discovery Grant (Yılmaz), and an NSERC Discovery Accelerator Award (Yılmaz).

## References

- 1.R. Baraniuk, M. Davenport, R. DeVore, M. Wakin, A simple proof of the restricted isometry property for random matrices,
*Constr. Approx.***28**(3), 253–263 (2008). MathSciNetzbMATHCrossRefGoogle Scholar - 2.J.J. Benedetto, A.M. Powell, Ö. Yılmaz, Second order sigma–delta (
*ΣΔ*) quantization of finite frame expansions,*Appl. Comput. Harmon. Anal.***20**, 126–148 (2006). MathSciNetzbMATHCrossRefGoogle Scholar - 3.J.J. Benedetto, A.M. Powell, Ö. Yılmaz, Sigma–delta (
*ΣΔ*) quantization and finite frames,*IEEE Trans. Inf. Theory***52**(5), 1990–2005 (2006). CrossRefGoogle Scholar - 4.R. Bhatia,
*Matrix Analysis*. Graduate Texts in Mathematics, vol. 169 (Springer, New York, 1997). CrossRefGoogle Scholar - 5.J. Blum, M. Lammers, A.M. Powell, Ö. Yılmaz, Sobolev duals in frame theory and sigma–delta quantization,
*J. Fourier Anal. Appl.***16**(3), 365–381 (2010). MathSciNetzbMATHCrossRefGoogle Scholar - 6.B.G. Bodmann, V.I. Paulsen, S.A. Abdulbaki, Smooth frame-path termination for higher order sigma–delta quantization,
*J. Fourier Anal. Appl.***13**(3), 285–307 (2007). MathSciNetzbMATHCrossRefGoogle Scholar - 7.P. Boufounos, R.G. Baraniuk, 1-bit compressive sensing. In:
*42nd Annual Conference on Information Sciences and Systems (CISS)*, pp. 19–21. Google Scholar - 8.P. Boufounos, R.G. Baraniuk, Sigma delta quantization for compressive sensing, in
*Wavelets XII*, ed. by D. Van de Ville, V.K. Goyal, M. Papadakis. Proceedings of SPIE, vol. 6701 (SPIE, Bellingham, 2007). Article CID 670104. CrossRefGoogle Scholar - 9.E.J. Candès, Compressive sampling, in
*International Congress of Mathematicians*, vol. III (Eur. Math. Soc., Zürich, 2006), pp. 1433–1452. Google Scholar - 10.E.J. Candès, J. Romberg, T. Tao, Signal recovery from incomplete and inaccurate measurements,
*Commun. Pure Appl. Math.***59**(8), 1207–1223 (2005). CrossRefGoogle Scholar - 11.E.J. Candès, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,
*IEEE Trans. Inf. Theory***52**(2), 489–509 (2006). zbMATHCrossRefGoogle Scholar - 12.A. Cohen, W. Dahmen, R. DeVore, Compressed sensing and best
*k*-term approximation,*J. Am. Math. Soc.***22**(1), 211–231 (2009). MathSciNetzbMATHCrossRefGoogle Scholar - 13.W. Dai, O. Milenkovic, Information theoretical and algorithmic approaches to quantized compressive sensing,
*IEEE Trans. Commun.***59**(7), 1857–1866 (2011). MathSciNetCrossRefGoogle Scholar - 14.I. Daubechies, R. DeVore, Approximating a bandlimited function using very coarsely quantized data: a family of stable sigma–delta modulators of arbitrary order,
*Ann. Math.***158**(2), 679–710 (2003). MathSciNetzbMATHCrossRefGoogle Scholar - 15.D.L. Donoho, Compressed sensing,
*IEEE Trans. Inf. Theory***52**(4), 1289–1306 (2006). MathSciNetCrossRefGoogle Scholar - 16.D.L. Donoho, For most large underdetermined systems of equations, the minimal l1-norm near-solution approximates the sparsest near-solution,
*Commun. Pure Appl. Math.***59**(7), 907–934 (2006). MathSciNetCrossRefGoogle Scholar - 17.A.K. Fletcher, S. Rangan, V.K. Goyal, Necessary and sufficient conditions for sparsity pattern recovery,
*IEEE Trans. Inf. Theory***55**(2009), 5758–5772 (2009). MathSciNetCrossRefGoogle Scholar - 18.V.K. Goyal, M. Vetterli, N.T. Thao, Quantized overcomplete expansions in ℝ
^{N}: analysis, synthesis, and algorithms,*IEEE Trans. Inf. Theory***44**(1), 16–31 (1998). MathSciNetzbMATHCrossRefGoogle Scholar - 19.V.K. Goyal, A.K. Fletcher, S. Rangan, Compressive sampling and lossy compression,
*IEEE Signal Process. Mag.***25**(2), 48–56 (2008). CrossRefGoogle Scholar - 20.C.S. Güntürk, One-bit sigma-delta quantization with exponential accuracy,
*Commun. Pure Appl. Math.***56**(11), 1608–1630 (2003). zbMATHCrossRefGoogle Scholar - 21.L. Jacques, D.K. Hammond, J.M. Fadili, Dequantizing compressed sensing: when oversampling and non-Gaussian constraints combine,
*IEEE Trans. Inf. Theory***57**(1), 559–571 (2011). MathSciNetCrossRefGoogle Scholar - 22.M.C. Lammers, A.M. Powell, Ö. Yılmaz, On quantization of finite frame expansions: sigma–delta schemes of arbitrary order,
*Proc. SPIE***6701**, 670108 (2007). CrossRefGoogle Scholar - 23.M. Lammers, A.M. Powell, Ö. Yılmaz, Alternative dual frames for digital-to-analog conversion in sigma–delta quantization,
*Adv. Comput. Math.***32**(1), 73–102 (2010). MathSciNetzbMATHCrossRefGoogle Scholar - 24.J.N. Laska, P.T. Boufounos, M.A. Davenport, R.G. Baraniuk, Democracy in action: quantization, saturation, and compressive sensing,
*Appl. Comput. Harmon. Anal.***31**(3), 429–443 (2011). MathSciNetzbMATHCrossRefGoogle Scholar - 25.G.G. Lorentz, M. von Golitschek, Y. Makovoz,
*Constructive Approximation*. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 304 (Springer, Berlin, 1996). Advanced problems. zbMATHCrossRefGoogle Scholar - 26.D. Needell, J. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples,
*Appl. Comput. Harmon. Anal.***26**(3), 301–321 (2009). MathSciNetzbMATHCrossRefGoogle Scholar - 27.D. Needell, R. Vershynin, Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit,
*Found. Comput. Math.***9**, 317–334 (2009). doi: 10.1007/s10208-008-9031-3. MathSciNetzbMATHCrossRefGoogle Scholar - 28.S.R. Norsworthy, R. Schreier, G.C. Temes (eds.),
*Delta–Sigma Data Converters*(IEEE Press, New York, 1997). Google Scholar - 29.R.J. Pai, Nonadaptive lossy encoding of sparse signals. PhD thesis, Massachusetts Institute of Technology (2006). Google Scholar
- 30.S.V. Parter, On the distribution of the singular values of Toeplitz matrices,
*Linear Algebra Appl.***80**, 115–130 (1986). MathSciNetzbMATHCrossRefGoogle Scholar - 31.M. Rudelson, R. Vershynin, Smallest singular value of a random rectangular matrix,
*Commun. Pure Appl. Math.***62**(12), 1595–1739 (2009). MathSciNetCrossRefGoogle Scholar - 32.G. Strang, The discrete cosine transform, SIAM Rev., 135–147 (1999). Google Scholar
- 33.J.Z. Sun, V.K. Goyal, Optimal quantization of random measurements in compressed sensing, in
*IEEE International Symposium on Information Theory*. ISIT 2009, (2009), pp. 6–10. CrossRefGoogle Scholar - 34.R. Tibshirani, Regression shrinkage and selection via the lasso,
*J. R. Stat. Soc., Ser. B, Stat. Methodol.***58**(1), 267–288 (1996). MathSciNetzbMATHGoogle Scholar - 35.J. von Neumann, Distribution of the ratio of the mean square successive difference to the variance,
*Ann. Math. Stat.***12**(4), 367–395 (1941). zbMATHCrossRefGoogle Scholar - 36.M.J. Wainwright, Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting,
*IEEE Trans. Inf. Theory***55**(12), 5728–5741 (2009). MathSciNetCrossRefGoogle Scholar - 37.M.J. Wainwright, Sharp thresholds for high-dimensional and noisy sparsity recovery using l1-constrained quadratic programming (Lasso),
*IEEE Trans. Inf. Theory***55**(5), 2183–2202 (2009). MathSciNetCrossRefGoogle Scholar - 38.A. Zymnis, S. Boyd, E. Candes, Compressed sensing with quantized measurements,
*IEEE Signal Process. Lett.***17**(2), 149 (2010). CrossRefGoogle Scholar