  We propose a new class of nonconvex penalty functions, 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 centered 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 algorithm for its
computation. For orthogonal design and independent responses, the resulting
thresholding rule enjoys near-minimax optimal risk performance, similar to the
adaptive lasso (Zou, 2006). A simulation study and real data analysis
demonstrate its effectiveness compared to some of the present methods that
provide sparse solutions in multivariate regression.
