Generalized linear model
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.
High-dimensional linear model.
formula_1
with formula_2 design matrix formula_3 (formula_4 vectors formula_5), formula_6 independent of formula_7 and unknown regression formula_8 vector formula_9.
The usual method to find the parameter is by Lasso:
formula_10
The de-sparsified lasso is a method modified from the Lasso estimator which fulfills the Karush–Kuhn–Tucker conditions is as follows:
formula_11
where formula_12 is an arbitrary matrix. The matrix formula_13 is generated using a surrogate inverse covariance matrix.
Generalized linear model.
Desparsifying formula_14-norm penalized estimators and corresponding theory can also be applied to models with convex loss functions such as generalized linear models.
Consider the following formula_15vectors of covariables formula_16 and univariate responses formula_17 for formula_18
we have a loss function 
formula_19
which is assumed to be strictly convex function in formula_20
The formula_14-norm regularized estimator is 
formula_22
Similarly, the Lasso for node wise regression with matrix input is defined as follows:
Denote by formula_23 a matrix which we want to approximately invert using nodewise lasso.
The de-sparsified formula_14-norm regularized estimator is as follows:
formula_25
where formula_26 denotes the formula_27th row of formula_23 without the diagonal element formula_29, and formula_30 is the sub matrix without the formula_27th row and formula_27th column.
References.
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