  We consider the problem of recovering a block (or group) sparse signal from
an underdetermined set of random linear measurements, which appear in
compressed sensing applications such as radar and imaging. Recent results of
Donoho, Johnstone, and Montanari have shown that approximate message passing
(AMP) in combination with Stein's shrinkage outperforms group LASSO for large
block sizes. In this paper, we prove that, for a fixed block size and in the
strong undersampling regime (i.e., having very few measurements compared to the
ambient dimension), AMP cannot improve upon group LASSO, thereby complementing
the results of Donoho et al.
