De-sparsified lasso

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De-sparsified lasso contributes to construct confidence intervals and statistical tests for single or low-dimensional components of a large parameter vector in high-dimensional model.[1]

1 High-dimensional linear model

with design matrix ( vectors ), independent of and unknown regression vector .

The usual method to find the parameter is by Lasso:

The de-sparsified lasso is a method modified from the Lasso estimator which fulfills the Karush-Kuhn-Tucker conditions[2] is as follows:

where is an arbitrary matrix. The matrix is generated using a surrogate inverse covariance matrix.

2 Generalized linear model

Desparsifying -norm penalized estimators and corresponding theory can also be applied to models with convex loss functions such as generalized linear models. p Consider the following vectors of covariables and univariate responses for which is assumed to be strictly convex function in

The -norm regularized estimator is

Similarly, the Lasso for node wise regression with matrix input is defined as follows: Denote by a matrix which we want to approximately invert using nodewise lasso.

The de-sparsified -norm regularized estimator is as follows:

where denotes the th row of without the diagonal element , and is the sub matrix without the th row and th column.

References

  1. ^ GEER, SARA VAN DE; BUHLMANN, PETER; RITOV, YA' ACOV; DEZEURE, RUBEN (2014). "ON ASYMPTOTICALLY OPTIMAL CONFIDENCE REGIONS AND TESTS FOR HIGH-DIMENSIONAL MODELS". The Annals of Statistics. 42: 1162–1202. arXiv:1303.0518Freely accessible. doi:10.1214/14-AOS1221. 
  2. ^ Tibshirani, Ryan; Gordon, Geoff. "Karush-Kuhn-Tucker conditions" (PDF). 
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