Error Bounds for Compressed Sensing Algorithms With Group Sparsity: A Unified Approach

In compressed sensing, in order to recover a sparse or nearly sparse vector from possibly noisy measurements, the most popular approach is $\ell1$-norm minimization. Upper bounds for the $\ell2$- norm of the error between the true and estimated vectors are given in [1] and reviewed in [2], while bounds for the $\ell1$-norm are given in [3]. When the unknown vector is not conventionally sparse but is "group sparse" instead, a variety of alternatives to the $\ell1$-norm have been proposed in the literature, including the group LASSO, sparse group LASSO, and group LASSO with tree structured overlapping groups. However, no error bounds are available for any of these modified objective functions. In the present paper, a unified approach is presented for deriving upper bounds on the error between the true vector and its approximation, based on the notion of decomposable and $\gamma$-decomposable norms. The bounds presented cover all of the norms mentioned above, and also provide a guideline for choosing norms in future to accommodate alternate forms of sparsity.