  In compressed sensing, in order to recover a sparse or nearly sparse vector
from possibly noisy measurements, the most popular approach is $\ell_1$-norm
minimization. Upper bounds for the $\ell_2$- norm of the error between the true
and estimated vectors are given in [1] and reviewed in [2], while bounds for
the $\ell_1$-norm are given in [3]. When the unknown vector is not
conventionally sparse but is "group sparse" instead, a variety of alternatives
to the $\ell_1$-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.
