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]

High-dimensional linear model

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  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.

Generalized linear model

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Desparsifying  -norm penalized estimators and corresponding theory can also be applied to models with convex loss functions such as generalized linear models.

Consider the following  vectors of covariables   and univariate responses   for  

we have a loss function   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

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  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 (3): 1162–1202. arXiv:1303.0518. doi:10.1214/14-AOS1221. S2CID 9663766.
  2. ^ Tibshirani, Ryan; Gordon, Geoff. "Karush-Kuhn-Tucker conditions" (PDF).