De-sparsified lasso
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.