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Sparsity Oracle Inequalities for Lasso and Dantzig Selector in High-Dimensional Nonparametric Regression

Shiqing Wang, Yan Shi, Limin Su

Abstract


Regularity conditions, such as the incoherence condition, restricted isometry property, compatibility condition and restricted eigenvalue assumption, play a pivotal role in high-dimensional regression and compressed sensing. Under these conditions, some interesting results for the Lasso and Dantzig selectors are derived. In this paper, we propose a modification of the compatibility condition, which is called modified compatibility condition. We show the oracle inequalities under the new condition and the methods which avoid using the sparsity condition. As a comparison with the results by Bickel et al. (2009) in high-dimensional nonparametric regression, more precise oracle inequalities for the prediction risk and bounds on the estimation loss are derived when the number of variables can be much larger than the sample size.

Keywords


Nonparametric regression, sparsity, Lasso, Dantzig selector, oracle inequality, prediction risk, estimation loss.

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