Backfitting algorithms for total-variation and empirical-norm penalized additive modelling with high-dimensional data

Additive modelling is useful in studying a non-linear relationship between a response and covariates. We develop backfitting algorithms to implement a doubly penalized method using total-variation and empirical-norm penalties with high-dimensional data. Use of the total-variation penalty leads to an automatic selection of knots for each component function, whereas use of the empirical-norm penalty can result in zero solutions for component functions and hence facilitates component selection in high-dimensional settings. For a backfitting cycle, each component function is updated by thresholding a solution to a Lasso problem, which is computed using an active-set (AS) descent method. Screening rules are also derived to determine zero solutions without solving the Lasso problem directly. We present numerical experiments to demonstrate the effectiveness of the proposed algorithms for linear and logistic additive modelling. (c) 2018 John Wiley & Sons, Ltd.