Asymptotic Confidence Regions for High-dimensional Structured Sparsity
This work addresses the need for reliable uncertainty quantification in high-dimensional statistical models with structured sparsity, which is incremental as it builds on existing desparsification methods.
The paper tackles the problem of constructing confidence regions for high-dimensional linear regression with structured sparsity, proposing two frameworks for pointwise and group confidence sets that incorporate prior knowledge about coefficient organization, and demonstrates their asymptotic behavior and differences through simulations.
In the setting of high-dimensional linear regression models, we propose two frameworks for constructing pointwise and group confidence sets for penalized estimators which incorporate prior knowledge about the organization of the non-zero coefficients. This is done by desparsifying the estimator as in van de Geer et al. [18] and van de Geer and Stucky [17], then using an appropriate estimator for the precision matrix $Θ$. In order to estimate the precision matrix a corresponding structured matrix norm penalty has to be introduced. After normalization the result is an asymptotic pivot. The asymptotic behavior is studied and simulations are added to study the differences between the two schemes.