Adaptive Dantzig density estimation

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 47, Issue 1, 2011, Pages 43-74

  Bertin, K ^a   Le Pennec, E ^b   Rivoirard, V ^c,d  

^a UNIVERSIDAD DE VALPARAÍSO   (Chile)

^b UNIVERSITÉ PARIS DESCARTES   (France)

^c UNIVERSITÉ PARIS SUD   (France)

^d ECOLE NORMALE SUPÉRIEURE   (France)

Author keywords

Calibration; Concentration inequalities; Dantzig estimate; Density estimation; Dictionary; Lasso estimate; Oracle inequalities; Sparsity

Indexed keywords

EID: 79251474881     PISSN: 02460203     EISSN: 02460203     Source Type: Journal    
DOI: 10.1214/09-AIHP351     Document Type: Article

References (32)

1. S. Arlot and P. Massart. Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res. 10 (2009) 245-279.
2. M. S. Asif and J. Romberg. Dantzig selector homotopy with dynamic measurements. In Proceedings of SPIE Computational Imaging VII 7246 (2009) 72460E.
3. P. Bickel, Y. Ritov and A. Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. Ann. Statist. 37 (2009) 1705-1732. MR2533469
4. 2-loss, 2008. Available at arXiv 0808.1416.
5. L. Birgé and P. Massart. Minimal penalties for Gaussian model selection. Probab. Theory Related. Fields 138 (2007) 33-73. MR2288064
6. 1 penalized least squares. In Learning Theory 379-391. Lecture Notes in Comput. Sci. 4005. Springer, Berlin, 2006. MR2280619
7. 1 penalties. Learning Theory 530-543. Lecture Notes in Comput. Sci. 4539. Springer, Berlin, 2007. MR2397610
8. F. Bunea, A. Tsybakov and M. Wegkamp. Sparsity oracle inequalities for the LASSO. Electron. J. Statist. 1 (2007) 169-194. MR2312149
9. F. Bunea, A. Tsybakov and M. Wegkamp. Aggregation for Gaussian regression. Ann. Statist. 35 (2007) 1674-1697. MR2351101
10. F. Bunea, A. Tsybakov, M. Wegkamp and A. Barbu. Spades and mixture models. Ann. Statist. (2010). To appear. Available at arXiv 0901.2044.
11. F. Bunea. Consistent selection via the Lasso for high dimensional approximating regression models. In Pushing the Limits of Contemporary Statistics: Cartributions in Honor of J. K. Ghosh 122-137. Inst. Math. Stat. Collect 3. IMS, Beachwood, OH, 2008. MR2459221
12. 1 minimization. Ann. Statist. 37 (2009) 2145-2177. MR2543688
13. E. Candès and T. Tao. The Dantzig selector: Statistical estimation when p is much larger than n. Ann. Statist. 35 (2007) 2313-2351. MR2382644
14. D. Chen, D. Donoho and M. Saunders. Atomic decomposition by basis pursuit. SIAMRev. 43 (2001) 129-159. MR1854649
15. D. Donoho, M. Elad and V. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory 52 (2006) 6-18. MR2237332
16. D. Donoho and I. Johnstone. Ideal spatial adaptation via wavelet shrinkage. Biometrika 81 (1994) 425-455. MR1311089
17. B. Efron, T. Hastie, I. Johnstone and R. Tibshirani. Least angle regression. Ann. Statist. 32 (2004) 407-499. MR2060166
18. A. Juditsky and S. Lambert-Lacroix. On minimax density estimation on ℝ. Bernoulli 10 (2004) 187-220. MR2046772
19. K. Knight and W Fu. Asymptotics for Lasso-type estimators. Ann. Statist. 28 (2000) 1356-1378. MR1805787
20. K. Lounici. Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators. Electron. J. Stat. 2 (2008) 90-102. MR2386087
21. P. Massart. Concentration inequalities and model selection. Lecture Notes in Math. 1896. Springer, Berlin. Lectures from the 33rd Summer School on Probability Theory held in Saint-Flour July 6-23 2003, 2007. MR2319879
22. N. Meinshausen and P. Buhlmann. High-dimensional graphs and variable selection with the Lasso. Ann. Statist. 34 (2006) 1436-1462. MR2278363
23. N. Meinshausen and B. Yu. Lasso-type recovery of sparse representations for high-dimensional data. Ann. Statist. 37 (2009) 246-270. MR2488351
24. M. Osborne, B. Presnell and B. Turlach. On the Lasso and its dual. J. Comput. Graph. Statist. 9 (2000) 319-337. MR1822089
25. M. Osborne, B. Presnell and B. Turlach. A new approach to variable selection in least squares problems. IMA J. Numer. Anal. 20 (2000) 389-404. MR1773265
26. P. Reynaud-Bouret and V. Rivoirard. Near optimal thresholding estimation of a Poisson intensity on the real line. Electron. J. Statist. 4 (2010) 172-238.
27. P. Reynaud-Bouret, V. Rivoirard and C. Tuleau. Adaptive density estimation: A curse of support? 2009. Available at arXiv 0907.1794.
28. R. Tibshirani. Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58 (1996) 267-288. MR1379242
29. S. van de Geer. High-dimensional generalized linear models and the Lasso. Ann. Statist. 36 (2008) 614-645. MR2396809
30. B. Yu and P. Zhao. On model selection consistency of Lasso estimators. J. Mach. Learn. Res. 7 (2006) 2541-2567. MR2274449
31. C. Zhang and J. Huang. The sparsity and bias of the Lasso selection in high-dimensional linear regression. Ann. Statist. 36 (2008) 1567-1594. MR2435448
32. H. Zou. The adaptive Lasso and its oracle properties. J. Amer. Statist. Assoc. 101 (2006) 1418-1429. MR2279469