Worst possible sub-directions in high-dimensional models

JOURNAL OF MULTIVARIATE ANALYSIS

卷:146

Article

van de Geer, Sara¹ 

[1] Swiss Fed Inst Technol, Seminar Stat, Zurich, Switzerland

关键词: De-sparsifying;    Graphical Lasso;    Irrepresentable condition;    Lasso;    Oracle rates;    Sub-direction;   

DOI  :  10.1016/j.jmva.2015.09.018

来源: Elsevier

【 摘 要 】

We examine the rate of convergence of the Lasso estimator of lower dimensional components of the high-dimensional parameter. Under bounds on the l(1)-norm on the worst possible sub-direction these rates are of order root vertical bar J vertical bar log p/n where p is the total number of parameters, n is the number of observations and J subset of {1, ...., p) represents a subset of the parameters. We also derive rates in sup-norm in terms of the rate of convergence in l(1)-norm. The irrepresentable condition on a set J requires that the l(1)-norm of the worst possible sub-direction is sufficiently smaller than one. In that case sharp oracle results can be obtained. Moreover, if the coefficients in J are small enough the Lasso will put these coefficients to zero. By de-sparsifying one obtains fast rates in supremum norm without conditions on the worst possible-sub-direction. The results are extended to M-estimation with l(1)-penalty for generalized linear models and exponential families. For the graphical Lasso this leads to an extension of known results to the case where the precision matrix is only approximately sparse. The bounds we provide are non-asymptotic but we also present asymptotic formulations for ease of interpretation. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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