Open Access Open Access  Restricted Access Subscription or Fee Access

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

Shiqing Wang, Yan Shi, Limin Su


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.


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

Full Text:


Disclaimer/Regarding indexing issue:

We have provided the online access of all issues and papers to the indexing agencies (as given on journal web site). It’s depend on indexing agencies when, how and what manner they can index or not. Hence, we like to inform that on the basis of earlier indexing, we can’t predict the today or future indexing policy of third party (i.e. indexing agencies) as they have right to discontinue any journal at any time without prior information to the journal. So, please neither sends any question nor expects any answer from us on the behalf of third party i.e. indexing agencies.Hence, we will not issue any certificate or letter for indexing issue. Our role is just to provide the online access to them. So we do properly this and one can visit indexing agencies website to get the authentic information. Also: DOI is paid service which provided by a third party. We never mentioned that we go for this for our any journal. However, journal have no objection if author go directly for this paid DOI service.