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.
[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.
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.
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