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Sparse Recovery Approaches

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Book cover Sparse Representation, Modeling and Learning in Visual Recognition

Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

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Abstract

In this chapter, some concepts of sparse recovery approach from five branches are outlined. First, it outlines the convex relaxation method to recover the sparse signal which includes the linear programming method, second-order cone programming method, -homotopy method, and elastic net. Second, it outlines some classic greedy algorithms such as, MP, OMP, CoSaMP, and iterated hard thresholding methods. Third, it outlines the sparse signal recovery from Baysian aspect such as sparse relevance vector machine and sparse Bayesian learning. Fourth, it outlines the -norm gradient minimization method, its formulation, solution, and application. Lastly, it outlines the sparse feature projection approach method.

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Notes

  1. 1.

    http://www-fp.mcs.anl.gov/otc/Guide/SoftwareGuide/Blurbs/bqpd.html.

References

  1. Baraniuk, R.G., Cevher, V., Duarte, M.F., Hegde, C.: Model-based compressive sensing. IEEE Trans. Inf. Theory 56(4), 1982–2001 (2010)

    Article  MathSciNet  Google Scholar 

  2. Blumensath, T., Davies, M.E.: Gradient pursuits. IEEE Trans. Signal Process. 56(6), 2370–2382 (2008)

    Article  MathSciNet  Google Scholar 

  3. Blumensath, T., Davies, M.E.: Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14(5–6), 629–654 (2008)

    Google Scholar 

  4. Blumensath, T., Davies, M.E.: Iterative hard thresholding for compressed sensing. Appl. Comput. Harmon. Anal. 27(3), 265–274 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Blumensath, T., Davies, M.E., et al.: How to use the iterative hard thresholding algorithm. In: Signal Processing with Adaptive Sparse Structured Representations (2009)

    Google Scholar 

  6. Bo, L., Ren, X., Fox, D.: Multipath sparse coding using hierarchical matching pursuit. In: IEEE CVPR (2013)

    Google Scholar 

  7. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  8. Bredies, K., Lorenz, D.A.: Iterated hard shrinkage for minimization problems with sparsity constraints. SIAM J. Sci. Comput. 30(2), 657–683 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Candes, E., Romberg, J.: l1-magic: Recovery of sparse signals via convex programming. http://www.acm.caltech.edu/l1magic/downloads/l1magic.pdf 4 (2005)

  10. Candès, E.J., Romberg, J., Tao, T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489–509 (2006)

    Article  MATH  Google Scholar 

  11. Candes, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Chen, S., Gunn, S.R., Harris, C.J.: The relevance vector machine technique for channel equalization application. IEEE Trans. Neural Netw. 12(6), 1529–1532 (2001)

    Article  Google Scholar 

  13. Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20(1), 33–61 (1998)

    Article  MathSciNet  Google Scholar 

  14. Cheng, H., Zheng, N., Sun, C., Wetering, H.: Boosted Gabor features applied to vehicle detection. In: IEEE ICPR (2006)

    Google Scholar 

  15. Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57(11), 1413–1457 (2004)

    Article  MATH  Google Scholar 

  16. Donoho, D.L., Elad, M.: Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization. Proc. Natl. Acad. Sci. 100(5), 2197–2202 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Donoho, D.L., Huo, X.: Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inf. Theory 47(7), 2845–2862 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Donoho, D.L., Tsaig, Y.: Fast Solution of l1-Norm Minimization Problems When the Solution May Be Sparse. Department of Statistics, Stanford University (2006)

    Google Scholar 

  19. Donoho, D.L., Tsaig, Y., Drori, I., Starck, J.L.: Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inf. Theory 58(2), 1094–1121 (2012)

    Article  MathSciNet  Google Scholar 

  20. Elad, M., Matalon, B., Zibulevsky, M.: Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization. Appl. Comput. Harmon. Anal. 23(3), 346–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Eldar, Y.C., Kutyniok, G.: Compressed Sensing: Theory and Applications. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  22. Farbman, Z., Fattal, R., Lischinski, D., Szeliski, R.: Edge-preserving decompositions for multi-scale tone and detail manipulation. In: ACM Transactions on Graphics (2008)

    Google Scholar 

  23. Figueiredo, M.A., Nowak, R.D., Wright, S.J.: Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J. Sel. Top. Signal Process. 1(4), 586–597 (2007)

    Article  Google Scholar 

  24. Fletcher, R.: Practical Methods of Optimization, 2nd edn. Wiley, Chichester (2004)

    Google Scholar 

  25. Gale, D.: Linear programming and the simplex method. Not. AMS 54(3), 364–369 (2007)

    MATH  MathSciNet  Google Scholar 

  26. Gilbert, A.C., Guha, S., Indyk, P., Muthukrishnan, S., Strauss, M.: Near-optimal sparse Fourier representations via sampling. In: ACM Symposium on Theory of Computing (2002)

    Google Scholar 

  27. Gilbert, A.C., Muthukrishnan, S., Strauss, M.: Improved time bounds for near-optimal sparse Fourier representations. In: Optics & Photonics (2005)

    Google Scholar 

  28. Hoyer, P.O.: Non-negative matrix factorization with sparseness constraints. J. Mach. Learn. Res. 5, 1457–1469 (2004)

    MATH  MathSciNet  Google Scholar 

  29. Huang, H., Makur, A.: Backtracking-based matching pursuit method for sparse signal reconstruction. IEEE Signal Process. Lett. 18(7), 391–394 (2011)

    Article  Google Scholar 

  30. Ji, S., Xue, Y., Carin, L.: Bayesian compressive sensing. IEEE Trans. Signal Process. 56(6), 2346–2356 (2008)

    Article  MathSciNet  Google Scholar 

  31. Koh, K., Kim, S.J., Boyd, S.: An interior-point method for large-scale? 1-regularized logistic regression. J. Mach. Learn. Res. 1(4), 606–617 (2007)

    MathSciNet  Google Scholar 

  32. Lange, K., Hunter, D.R., Yang, I.: Optimization transfer using surrogate objective functions. J. Comput. Graph. Stat. 9(1), 1–20 (2000)

    MathSciNet  Google Scholar 

  33. Li, Q., Lin, N.: The bayesian elastic net. Bayesian Anal. 5(1), 151–170 (2010)

    Article  MathSciNet  Google Scholar 

  34. Needell, D., Tropp, J.A.: Cosamp: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harmon. Anal. 26(3), 301–321 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  35. Needell, D., Vershynin, R.: Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math. 9(3), 317–334 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  36. Needell, D., Vershynin, R.: Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit. IEEE Sel. Top. Signal Process. 4(2), 310–316 (2010)

    Article  Google Scholar 

  37. Nesterov, Y., Nemirovskii, A.S., Ye, Y.: Interior-point polynomial algorithms in convex programming. SIAM 13, 405 (1994)

    Google Scholar 

  38. Osborne, M.R., Presnell, B., Turlach, B.A.: A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 20(3), 389–403 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Osborne, M.R., Presnell, B., Turlach, B.A.: On the Lasso and its dual. J. Comput. Graph. Stat. 9(2), 319–337 (2000)

    MathSciNet  Google Scholar 

  40. Pati, Y.C., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In: IEEE Asilomar Conference on Signals, Systems and Computers (1993)

    Google Scholar 

  41. Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge. http://www.gaussianprocess.org/gpml/ (2006)

  42. Rudin, L.I., Osher, S., Fatemi, E.: Nonlinear total variation based noise removal algorithms. Phys. D: Nonlinear Phenom. 60(1), 259–268 (1992)

    Article  MATH  Google Scholar 

  43. Sun, C., Bebis, G., Miller, R.: On-road vehicle detection using evolutionary Gabor filter optimization. IEEE Trans. Intell. Transp. Syst. 6(2), 125–137 (2005)

    Article  Google Scholar 

  44. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. 58(1), 267–288 (1996)

    MATH  MathSciNet  Google Scholar 

  45. Tipping, M.E.: Sparse bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1, 211–244 (2001)

    MATH  MathSciNet  Google Scholar 

  46. Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: IEEE ICCV (1998)

    Google Scholar 

  47. Tropp, J.A., Gilbert, A.C.: Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  48. Weldon, T.P., Higgins, W.E., Dunn, D.F.: Efficient Gabor filter design for texture segmentation. Pattern Recognit. 29(12), 2005–2015 (1996)

    Article  Google Scholar 

  49. Wipf, D.P., Rao, B.D.: Sparse Bayesian learning for basis selection. IEEE Trans. Signal Process. 52(8), 2153–2164 (2004)

    Article  MathSciNet  Google Scholar 

  50. Xu, L., Lu, C., Xu, Y., Jia, J.: Image smoothing via l 0 gradient minimization. ACM Trans. Graph. 30(6), 174 (2011)

    Google Scholar 

  51. Yang, A.Y., Sastry, S.S., Ganesh, A., Ma, Y.: Fast \(\ell _1\)-minimization algorithms and an application in robust face recognition: a review. In: IEEE ICIP (2010)

    Google Scholar 

  52. Yu, S., Shan, S., Chen, X., Gao, W.: Hierarchical ensemble of global and local classifiers for face recognition. In: IEEE CVPR (2007)

    Google Scholar 

  53. Zou, H., Hastie, T.: Regularization and variable selection via the elastic net. J. R. Stat. Soc. 67(2), 301–320 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Hong Cheng

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  1. Hong Cheng
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Correspondence to Hong Cheng .

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Cheng, H. (2015). Sparse Recovery Approaches. In: Sparse Representation, Modeling and Learning in Visual Recognition. Advances in Computer Vision and Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-4471-6714-3_3

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  • DOI: https://doi.org/10.1007/978-1-4471-6714-3_3

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  • Publisher Name: Springer, London

  • Print ISBN: 978-1-4471-6713-6

  • Online ISBN: 978-1-4471-6714-3

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