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Optimal D-RIP bounds in compressed sensing

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Abstract

This paper establishes new bounds on the restricted isometry constants with coherent tight frames in compressed sensing. It is shown that if the sensing matrix A satisfies the D-RIP condition δ k < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), then all signals f with D*f are k-sparse can be recovered exactly via the constrained ℓ 1 minimization based on y = Af, where D* is the conjugate transpose of a tight frame D. These bounds are sharp when D is an identity matrix, see Cai and Zhang’s work. These bounds are greatly improved comparing to the condition δ k < 0.307 or δ 2k < 0.4931. Besides, if δ k < 1/3 or \(\delta _{2k} < \sqrt 2 /2\), the signals can also be stably reconstructed in the noisy cases.

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References

  1. Aldroubi, A., Chen, X., Powell, A. M.: Perturbations of measurement matrices and dictionaries in compressed sensing. Appl. Comput. Harmon. Anal., 33, 282–291 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blanchard, J. D., Cartis, C., Tanner J.: Compressed sensing: How sharp is the RIP? SIAM Rev., 53, 105–125 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baraniuk, R., Davenport, M., DeVore, R., et al.: A simple proof of the restricted isometry property for random matrices. Constr. Approx., 28, 253–263 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cai, T., Wang, L., Xu, G.: Shifting inequality and recovery of sparse signals. IEEE Trans. Signal Process., 58, 1300–1308 (2010)

    Article  MathSciNet  Google Scholar 

  5. Cai, T., Wang, L., Xu, G.: New bounds for restricted isometry constants. IEEE Trans. Inform. Theory, 56, 4388–4394 (2010)

    Article  MathSciNet  Google Scholar 

  6. Cai, T., Xu, G., Zhang, J.: On recovery of sparse signals via ℓ 1 minimization. IEEE Trans. Infrom. Theory, 55, 3388–3397 (2009)

    Article  MathSciNet  Google Scholar 

  7. Cai, T., Zhang, A.: Sparse representation of a polytope and recovery of sparse signals and low-rank matrices. IEEE Trans. Inform. Theory, 60, 122–132 (2014)

    Article  MathSciNet  Google Scholar 

  8. Cai, T., Zhang, A.: Sharp RIP bound for sparse signal and low-rank matrix recovery. Appl. Comput. Harmon. Anal., 35, 74–93 (2012)

    Article  MathSciNet  Google Scholar 

  9. Candès, E. J.: The restricted isometry property and its implications for compressed sensing. C. R. Math. Acad. Sci. Paris, Serie I, 346, 589–592 (2008)

    Article  MATH  Google Scholar 

  10. Candès, E. J., Eldar, Y. C., Needell, D.: Compressed sensing with coherent and redundant dictionaries. Appl. Comput. Harmon. Anal., 31, 59–73 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Candès, E. J., Romberg, J., Tao, T.: Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 52, 489–509 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Candès, E. J., Tao, T.: Decoding by linear programming. IEEE Trans. Inform. Theory, 51, 4203–4215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Candès, E. J., Tao, T.: The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist., 35, 2313–2351 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Davies, M. E., Gribonval, R.: Restricted isometry properties where ℓ p sparse recovery can fail for 0 < p ≤ 1. IEEE Trans. Inform. Theory, 55, 2203–2214 (2010)

    Article  MathSciNet  Google Scholar 

  15. Donoho, D. L.: Compressed sensing. IEEE Trans. Inform. Theory, 52, 1289–1306 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Donoho, D. L.: For most large underdetermined systems of linear equations the minimal ℓ 1 solution is also the sparsest solution. Comm. Pure Appl. Math., 59, 797–829 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Foucart, S., Lai, M. J.: Sparsest solutions of underdetermined linear systems via ℓ q minimization for 0 < q ≤ 1. Appl. Comput. Harmon. Anal., 26, 395–407 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, S., Lin, J.: Compressed sensing with coherent tight frames via ℓ q-minimization for 0 < q ≤ 1. Inverse Probl. Imaging, 8, 761–777 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, J., Li, S., Shen, Y.: New bounds for restricted isometry constants with coherent tight frames. IEEE Trans. Signal Process., 61, 611–621 (2013)

    Article  MathSciNet  Google Scholar 

  20. Liu, Y., Mi, T., Li, S.: Compressed sensing with general frames via optimal-dual-based ℓ 1-analysis. IEEE Trans. Inform. Theory, 58, 4201–4214 (2012)

    Article  MathSciNet  Google Scholar 

  21. Mo, Q., Li, S.: New bounds on the restricted isometry constant δ 2k . Appl. Comput. Harmon. Anal., 31, 460–468 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rauhut, H., Schnass, K., Vandergheynst, P.: Compressed sensing and redundant dictionaries. IEEE Trans. Inform. Theory, 54, 2210–2219 (2008)

    Article  MathSciNet  Google Scholar 

  23. Saab, R., Yılmaz, Ö.: Sparse recovery by non-convex optimization-instance optimality. Appl. Comput. Harmon. Anal., 29, 30–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Zhejiang University, Hangzhou, 310027, P. R. China

    Rui Zhang & Song Li

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  1. Rui Zhang
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  2. Song Li
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Correspondence to Rui Zhang.

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Supported by NSFC (Grant No. 11171299)

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Zhang, R., Li, S. Optimal D-RIP bounds in compressed sensing. Acta. Math. Sin.-English Ser. 31, 755–766 (2015). https://doi.org/10.1007/s10114-015-4234-4

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  • DOI: https://doi.org/10.1007/s10114-015-4234-4

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