Flexible Variable Selection for Recovering Sparsity in Nonadditive Nonparametric Models

Abstract: Variable selection for recovering sparsity in nonadditive nonparametric models has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this paper we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric multiple regression model. The advantages of our approach are that it can: (1) recover the sparsity, (2) automatically model unknown and complicated interactions, (3) connect with several existing approaches including linear nonnegative garrote, kernel learning and automatic relevant determinants (ARD), and (4) provide flexibility for both additive and nonadditive nonparametric models. Our approach may be viewed as a nonlinear version of a nonnegative garrote method. We model the smoothing function by a least squares kernel machine and construct the nonnegative garrote objective function as the function of the similarity matrix. Since the multiple regression similarity matrix can be written as an additive form of univariate similarity matrices corresponding to input variables, applying a sparse scale parameter on each univariate similarity matrix can reveal its relevance to the response variable. We also derive the asymptotic properties of our approach, and show that it provides a square root consistent estimator of the scale parameters. Furthermore, we prove that sparsistency is satisfied with consistent initial kernel function coefficients under certain conditions and give the necessary and sufficient conditions for sparsistency. An efficient coordinate descent/backfitting algorithm is developed. A resampling procedure for our variable selection methodology is also proposed to improve power.