Multi-Task Nonparametric Regression Under Joint Sparsity

This study investigates a multi-task estimation under joint sparsity. We consider estimating multiple functions when functions of interest share common sparsity patterns. An $\ell _{2}$ penalty is imposed to enforce common sparsity patterns across component functions. A non-asymptotic oracle inequality is established to illustrate a possible improvement of the estimation error bound achieved by the proposed pooled estimator in comparison with the usual projection estimator. The proposed method is implemented with the alternating direction method of multipliers algorithm. Numerical studies are conducted to complement the theoretical results. We apply the proposed method to the ozone data to illustrate a practical applicability. The numerical results show that the proposed method detects the underlying sparsity patterns, thereby providing a desirable estimator that significantly outperforms the projection estimator.