The Group Dantzig Selector
Han Liu, Jian Zhang, Xiaoye Jiang, Jun Liu ; JMLR W&CP 9:461-468, 2010.
We introduce a new method -- the group Dantzig selector -- for high dimensional sparse regression with group structure, which has a convincing theory about why utilizing the group structure can be beneficial. Under a group restricted isometry condition, we obtain a significantly improved nonasymptotic L2-norm bound over the basis pursuit or the Dantzig selector which ignores the group structure. To gain more insight, we also introduce a surprisingly simple and intuitive "sparsity oracle condition" to obtain a block L1-norm bound, which is easily accessible to a broad audience in machine learning community. Encouraging numerical results are also provided to support our theory.