Error Bounds for Compressed Sensing Algorithms With Group Sparsity: A Unified Approach
This work addresses a theoretical gap for researchers in signal processing and machine learning, providing a framework for error analysis in group sparsity settings, though it is incremental as it extends existing bounds to new norms.
The paper tackles the lack of error bounds for compressed sensing algorithms with group sparsity by presenting a unified approach to derive upper bounds on the error between true and estimated vectors, covering norms like group LASSO and sparse group LASSO.
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 $γ$-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.