Screening is the problem of estimating a superset of the set of non-zero entries in an unknown p-dimensional vector \beta* given n noisy observations. In the high-dimensional regime, where p > n, screening algorithms are useful for reducing the dimensionality of \beta*. This allows for using computationally challenging algorithms, such as cross-validation or stability selection, for estimating \beta* and/or the set of non-zero entries in \beta*. We propose a novel framework for screening, which we refer to as Multiple Grouping (MuG), that groups variables, performs variable selection over the groups, and repeats this process multiple number of times to estimate a sequence of sets that contains the non-zero entries in \beta*. Screening is done by taking an intersection of all these estimated sets. We show how MuG, when used in conjunction with the group Lasso estimator, can consistently perform screening and reduce the dimensionality of \beta*. The main advantage of using MuG in conjunction with the group Lasso is that it leads to a parameter-free screening algorithm so that screening can be done without using any tuning parameter. Our numerical simulations clearly show the merits of applying the MuG framework in practice.
['Divyanshu Vats']
Parameter-Free High-Dimensional Screening Using Multiple Grouping of Variables