The Dantzig selector and sparsity oracle inequalities

Bernoulli

Volumn 15, Issue 3, 2009, Pages 799-828

  Koltchinskii, Vladimir ^a  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Dantzig selector; Oracle inequalities; Regression; Sparsity

Indexed keywords

EID: 72249100613     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/09-BEJ187     Document Type: Article

References (16)

1. Bickel, P., Ritov, Y. and Tsybakov, A. (2009). Simultaneous analysis of LASSO and Dantzig selector. Ann. Statist. To appear.
2. Bobkov, S. and Houdré, C. (1997). Isoperimetric constants for product probability measures. Ann. Probab. 25 184-205. MR1428505
3. Bunea, F., Tsybakov, A. and Wegkamp, M. (2007). Sparsity oracle inequalities for the LASSO. Electron. J. Statist. 1 169-194. MR2312149
4. Candes, E., Romberg, J. and Tao, T. (2006). Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 489-509. MR2236170
5. Candes, E. and Tao, T. (2005). Decoding by linear programming. IEEE Trans. Inform. Theory 51 4203-4215. MR2243152
6. Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 2313-2351. MR2382644
7. de la Pena, V. and Giné, E. (1998). Decoupling: From Dependence to Independence. New York: Springer. MR1666908
8. 1-norm solution is also the sparsest solution. Commun. Pure Appl. Math. 59 797-829. MR2217606
9. 1-norm nearsolution approximates the sparsest near-solution. Commun. Pure Appl. Math. 59 907-934.
10. Koltchinskii, V. (2005). Model selection and aggregation in sparse classification problems. In Oberwolfach Reports: Meeting on Statistical and Probabilistic Methods of Model Selection, October 2005. European Mathematical Society Publishing House.
11. Koltchinskii, V. (2009). Sparsity in penalized empirical risk minimization. Ann. Inst. H. Poincaré Probab. Statist. 45 7-57.
12. Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. New York: Springer. MR1102015
13. Mendelson, S., Pajor, A. and Tomczak-Jaegermann, N. (2007). Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17 1248-1282. MR2373017
14. Rudelson, M. and Vershynin, R. (2005). Geometric approach to error correcting codes and reconstruction of signals. Int. Math. Res. Not. 64 4019-4041. MR2206919
15. van de Geer, S. (2008). High-dimensional generalized linear models and the Lasso. Ann. Statist. 36 614-645. MR2396809
16. van der Vaart, A. and Wellner, J. (1996). Weak Convergence and Empirical Processes. New York: Springer. MR1385671