Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 946241, 7 pages
http://dx.doi.org/10.1155/2014/946241
Research Article

Weaker Regularity Conditions and Sparse Recovery in High-Dimensional Regression

1College of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450045, China
2Institute of Environmental and Municipal Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450045, China

Received 27 October 2013; Accepted 7 July 2014; Published 17 July 2014

Academic Editor: Yuesheng Xu

Copyright © 2014 Shiqing Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society B, vol. 58, no. 1, pp. 267–288, 1996. View at Google Scholar · View at MathSciNet
  2. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Journal on Scientific Computing, vol. 20, no. 1, pp. 33–61, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. E. Candes and T. Tao, “The Dantzig selector: statistical estimation when p is much larger than n,” The Annals of Statistics, vol. 35, no. 6, pp. 2313–2351, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. E. Gautier and A. B. Tsybakov, “High-dimensional instrumental variables regression and confidence sets,” Working Paper, 2011. View at Google Scholar
  5. P. Bühlmann and S. van de Geer, Statistics for High-Dimensional Data, Springer Series in Statistics, Springer, New York, NY, USA, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. E. J. Candes, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp. 1207–1223, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. S. Q. Wang and L. M. Su, “The oracle inequalities on simultaneous Lasso and Dantzig selector in high-dimensional nonparametric regression,” Mathematical Problems in Engineering, vol. 2013, Article ID 571361, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. A. van de Geer and P. Buhlmann, “On the conditions used to prove oracle results for the Lasso,” Electronic Journal of Statistics, vol. 3, pp. 1360–1392, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Q. Wang and L. M. Su, “Recovery of high-dimensional spares signals via l1-minimization,” Journal of Applied Mathematics, vol. 2013, Article ID 636094, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. P. J. Bickel, Y. Ritov, and A. B. Tsybakov, “Simultaneous analysis of lasso and Dantzig selector,” The Annals of Statistics, vol. 37, no. 4, pp. 1705–1732, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. van de Geer, The Deterministic Lasso, Seminar für Statistik, Eidgenössische Technische Hochschule (ETH), Zürich, Switzerland, 2007.
  12. E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4203–4215, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Q. Wang and L. M. Su, “Simultaneous lasso and dantzig selector in high dimensional nonparametric regression,” International Journal of Applied Mathematics and Statistics, vol. 42, no. 12, pp. 103–118, 2013. View at Google Scholar · View at Scopus
  14. S. Q. Wang and L. M. Su, “New bounds of mutual incoherence property on sparse signals recovery,” International Journal of Applied Mathematics and Statistics, vol. 47, no. 17, pp. 462–477, 2013. View at Google Scholar
  15. P. Alquier and M. Hebiri, “Generalization of L1 constraints for high dimensional regression problems,” Statistics and Probability Letters, vol. 81, no. 12, pp. 1760–1765, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. P. Zhao and B. Yu, “On model selection consistency of Lasso,” The Journal of Machine Learning Research, vol. 7, no. 12, pp. 2541–2563, 2006. View at Google Scholar · View at MathSciNet · View at Scopus
  17. R. Adamczak, A. E. Litvak, A. Pajor, and N. Tomczak-Jaegermann, “Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling,” Constructive Approximation, vol. 34, no. 1, pp. 61–88, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, “Uniform uncertainty principle for Bernoulli and subgaussian ensembles,” Constructive Approximation, vol. 28, no. 3, pp. 277–289, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. R. G. Baraniuk, R. A. DeVore, and M. B. Davenport, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation, vol. 28, no. 3, pp. 253–263, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. V. Koltchinskii, “The Dantzig selector and sparsity oracle inequalities,” Bernoulli, vol. 15, no. 3, pp. 799–828, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. Y. de Castro, “A remark on the lasso and the Dantzig selector,” Statistics and Probability Letters, vol. 83, no. 1, pp. 304–314, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus