Non-convex penalized multitask regression using data depth-based penalties

We propose a new class of non-convex penalties based on data depth functions for multitask sparse penalized regression. These penalties quantify the relative position of rows of the coefficient matrix from a fixed distribution centred at the origin. We derive the theoretical properties of an approximate one-step sparse estimator of the coefficient matrix using local linear approximation of the penalty function and provide an algorithm for its computation. For the orthogonal design and independent responses, the resulting thresholding rule enjoys near-minimax optimal risk performance, similar to the adaptive lasso (Zou, H (2006), ‘The adaptive lasso and its oracle properties’, Journal of the American Statistical Association, 101, 1418–1429). A simulation study and real data analysis demonstrate its effectiveness compared with some of the present methods that provide sparse solutions in multitask regression.