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http://dx.doi.org/10.4134/JKMS.j180321

STABLE AND ROBUST ℓp-CONSTRAINED COMPRESSIVE SENSING RECOVERY VIA ROBUST WIDTH PROPERTY  

Yu, Jun (Department of Mathematics and Mathematical Statistics Umea University)
Zhou, Zhiyong (Department of Mathematics and Mathematical Statistics Umea University)
Publication Information
Journal of the Korean Mathematical Society / v.56, no.3, 2019 , pp. 689-701 More about this Journal
Abstract
We study the recovery results of ${\ell}_p$-constrained compressive sensing (CS) with $p{\geq}1$ via robust width property and determine conditions on the number of measurements for standard Gaussian matrices under which the property holds with high probability. Our paper extends the existing results in Cahill and Mixon from ${\ell}_2$-constrained CS to ${\ell}_p$-constrained case with $p{\geq}1$ and complements the recovery analysis for robust CS with ${\ell}_p$ loss function.
Keywords
compressive sensing; robust width property; robust null space property; restricted isometry property;
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1 Y. C. Eldar, P. Kuppinger, and H. Bolcskei, Block-sparse signals: uncertainty relations and efficient recovery, IEEE Trans. Signal Process. 58 (2010), no. 6, 3042-3054.   DOI
2 Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications, Cambridge University Press, 2012.
3 S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Applied and Numerical Harmonic Analysis, Birkhauser/Springer, New York, 2013.
4 M. P. Friedlander, H. Mansour, R. Saab, and O. Yilmaz, Recovering compressively sampled signals using partial support information, IEEE Trans. Inform. Theory 58 (2012), no. 2, 1122-1134.   DOI
5 L. Jacques, D. K. Hammond, and J. M. Fadili, Dequantizing compressed sensing: when oversampling and non-Gaussian constraints combine, IEEE Trans. Inform. Theory 57 (2011), no. 1, 559-571.   DOI
6 M. Kabanava, R. Kueng, H. Rauhut, and U. Terstiege, Stable low-rank matrix recovery via null space properties, Inf. Inference 5 (2016), no. 4, 405-441.   DOI
7 H. Mansour and R. Saab, Recovery analysis for weighted ${\ell}_1$-minimization using the null space property, Appl. Comput. Harmon. Anal. 43 (2017), no. 1, 23-38.   DOI
8 D. Needell and R.Ward, Near-optimal compressed sensing guarantees for total variation minimization, IEEE Trans. Image Process. 22 (2013), no. 10, 3941-3949.   DOI
9 D. Needell and R.Ward, Stable image reconstruction using total variation minimization, SIAM J. Imaging Sci. 6 (2013), no. 2, 1035-1058.   DOI
10 M. F. Duarte and Y. C. Eldar, Structured compressed sensing: from theory to applications, IEEE Trans. Signal Process. 59 (2011), no. 9, 4053-4085.   DOI
11 H. Rauhut and R. Ward, Interpolation via weighted ${\ell}_1$ minimization, Appl. Comput. Harmon. Anal. 40 (2016), no. 2, 321-351.   DOI
12 F. Wen, P. Liu, Y. Liu, R. C. Qiu, and W. Yu, Robust sparse recovery in impulsive noise via ${\ell}_p$-${\ell}_1$ optimization, IEEE Trans. Signal Process. 65 (2017), no. 1, 105-118.   DOI
13 A. A. Saleh, F. Alajaji, and W. Y. Chan, Compressed sensing with non-Gaussian noise and partial support information, IEEE Signal Processing Letters 22 (2015), no. 10, 1703-1707.   DOI
14 G. Tang and A. Nehorai, Performance analysis of sparse recovery based on constrained minimal singular values, IEEE Trans. Signal Process. 59 (2011), no. 12, 5734-5745.   DOI
15 R. Tibshirani, Regression shrinkage and selection via the lasso, J. Roy. Statist. Soc. Ser. B 58 (1996), no. 1, 267-288.
16 J. Cahill and D. G. Mixon, Robust width: A characterization of uniformly stable and robust compressed sensing, arXiv preprint arXiv:1408.4409, 2014.
17 H. Zhang and L. Cheng, On the constrained minimal singular values for sparse signal recovery, IEEE Signal Processing Letters 19 (2012), no. 8, 499-502.   DOI
18 Z. Zhou and J. Yu, Sparse recovery based on q-ratio constrained minimal singular values, Signal Processing 155 (2019), 247-258.   DOI
19 Z. Allen-Zhu, R. Gelashvili, and I. Razenshteyn, Restricted isometry property for general p-norms, IEEE Trans. Inform. Theory 62 (2016), no. 10, 5839-5854.   DOI
20 E. J. Candes, The restricted isometry property and its implications for compressed sensing, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 589-592.   DOI
21 E. J. Candes, J. K. Romberg, and T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59 (2006), no. 8, 1207-1223.   DOI
22 E. Candes and T. Tao, The Dantzig selector: statistical estimation when p is much larger than n, Ann. Statist. 35 (2007), no. 6, 2313-2351.   DOI
23 A. Cohen, W. Dahmen, and R. DeVore, Compressed sensing and best k-term approximation, J. Amer. Math. Soc. 22 (2009), no. 1, 211-231.   DOI
24 H. Zhang, On robust width property for Lasso and Dantzig selector, Commun. Math. Sci. 15 (2017), no. 8, 2387-2393.   DOI
25 D. L. Donoho, Compressed sensing, IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289-1306.   DOI
26 S. Dirksen, G. Lecue, and H. Rauhut, On the gap between restricted isometry properties and sparse recovery conditions, IEEE Trans. Inform. Theory 64 (2018), no. 8, 5478-5487.   DOI