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On the conditions used to prove oracle results for the Lasso. (English) Zbl 1327.62425

Summary: Oracle inequalities and variable selection properties for the Lasso in linear models have been established under a variety of different assumptions on the design matrix. We show in this paper how the different conditions and concepts relate to each other. The restricted eigenvalue condition [P. J. Bickel et al., Ann. Stat. 37, No. 4, 1705–1732 (2009; Zbl 1173.62022)] or the slightly weaker compatibility condition [the first author, The deterministic Lasso. Zürich: Seminar für Statistik, Eidgenössische Technische Hochschule (2007)] are sufficient for oracle results. We argue that both these conditions allow for a fairly general class of design matrices. Hence, optimality of the Lasso for prediction and estimation holds for more general situations than what it appears from coherence [F. Bunea et al., Lect. Notes Comput. Sci. 4539, 530–543 (2007; Zbl 1203.62053); Electron. J. Stat. 1, 169–194 (2007; Zbl 1146.62028)] or restricted isometry [E. J. Candès and T. Tao, IEEE Trans. Inf. Theory 51, No. 12, 4203–4215 (2005; Zbl 1264.94121)] assumptions.

MSC:

62J07 Ridge regression; shrinkage estimators (Lasso)
62C05 General considerations in statistical decision theory
62G05 Nonparametric estimation
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References:

[1] Bertsimas, and Tsitsiklis, (1997)., Introduction to linear optimization . Athena Scientific Belmont, MA.
[2] Bickel, Ritov, and Tsybakov, (2009). Simultaneous analysis of Lasso and Dantzig selector., Annals of Statistics 37 1705-1732. · Zbl 1173.62022
[3] Bunea, Tsybakov, and Wegkamp, (2007a). Aggregation for Gaussian regression., Annals of Statistics 35 1674. · Zbl 1209.62065
[4] Bunea, Tsybakov, and Wegkamp, (2007c). Sparsity oracle inequalities for the Lasso., Electronic Journal of Statistics 1 169-194. · Zbl 1146.62028
[5] Bunea, Tsybakov, and Wegkamp, (2007b). Sparse Density Estimation with, \ell 1 Penalties. In Learning Theory 20th Annual Conference on Learning Theory, COLT 2007, San Diego, CA, USA, June 13-15, 2007: Proceedings 530. Springer. · Zbl 1203.62053
[6] Cai, Wang, and Xu, (2009a). Shifting inequality and recovery of sparse signals., IEEE Transactions on Signal Processing, · Zbl 1392.94117
[7] Cai, Wang, and Xu, (2009b). Stable recovery of sparse signals and an oracle inequality., · Zbl 1366.94085
[8] Cai, Xu, and Zhang, (2009). On recovery of sparse signals via, \ell 1 minimization. IEEE Transactions on Information Theory 55 3388-3397. · Zbl 1367.94081
[9] Candès, and Plan, (2009). Near-ideal model selection by, \ell 1 minimization. Annals of Statistics 37 2145-2177. · Zbl 1173.62053
[10] Candès, and Tao, (2005). Decoding by linear programming., IEEE Transactions on Information Theory 51 4203-4215. · Zbl 1264.94121
[11] Candès, and Tao, (2007). The Dantzig selector: statistical estimation when p is much larger than n., Annals of Statistics 35 2313-2351. · Zbl 1139.62019
[12] Koltchinskii, (2009a). Sparsity in penalized empirical risk minimization., Annales de l’Institut Henri Poincaré, Probabilités et Statistiques 45 7-57. · Zbl 1168.62044
[13] Koltchinskii, (2009b). The Dantzig selector and sparsity oracle inequalities., Bernoulli 15 799-828. · Zbl 1452.62486
[14] Lounici, (2008). Sup-norm convergence rate and sign concentration property of Lasso and Dantzig estimators., Electronic Journal of Statistics 2 90-102. · Zbl 1306.62155
[15] Meinshausen, and Bühlmann, (2006). High-dimensional graphs and variable selection with the Lasso., Annals of Statistics 34 1436-1462. · Zbl 1113.62082
[16] Meinshausen, and Yu, (2009). Lasso-type recovery of sparse representations for high-dimensional data., Annals of Statistics 37 246-270. · Zbl 1155.62050
[17] Parter, (1961). Extreme eigenvalues of Toeplitz forms and applications to elliptic difference equations., Transactions of the American Mathematical Society 99 153-192. · Zbl 0099.32403
[18] van de Geer, (2007). The deterministic Lasso. In, JSM proceedings, (see also http://stat.ethz.ch/research/research_reports/2007/140) . American Statistical Association.
[19] van de Geer, (2008). High-dimensional generalized linear models and the Lasso., Annals of Statistics 36 614-645. · Zbl 1138.62323
[20] Wainwright, (2009). Sharp thresholds for high-dimensional and noisy sparsity recovery using, \ell 1 -constrained quadratic programming (Lasso). IEEE Transactions on Information Theory 55 2183-2202. · Zbl 1367.62220
[21] Zhang, and Huang, (2008). The sparsity and bias of the Lasso selection in high-dimensional linear regression., Annals of Statistics 36 1567-1594. · Zbl 1142.62044
[22] Zhang, (2009). Some sharp performance bounds for least squares regression with L1 regularization., Annals of Statistics 37 2109-2144. · Zbl 1173.62029
[23] Zhao, and Yu, (2006). On model selection consistency of Lasso., Journal of Machine Learning Research 7 2541-2563. · Zbl 1222.62008
[24] Zou, (2006). The adaptive Lasso and its oracle properties., Journal of the American Statistical Association 101 1418-1429. · Zbl 1171.62326
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