  We consider the problem of recovering an $N$-dimensional sparse vector
$\vm{x}$ from its linear transformation $\vm{y}=\vm{D} \vm{x}$ of $M(< N)$
dimension. Minimizing the $l_{1}$-norm of $\vm{x}$ under the constraint $\vm{y}
= \vm{D} \vm{x}$ is a standard approach for the recovery problem, and earlier
studies report that the critical condition for typically successful
$l_1$-recovery is universal over a variety of randomly constructed matrices
$\vm{D}$. For examining the extent of the universality, we focus on the case in
which $\vm{D}$ is provided by concatenating $\nb=N/M$ matrices $\vm{O}_{1},
\vm{O}_{2},..., \vm{O}_\nb$ drawn uniformly according to the Haar measure on
the $M \times M$ orthogonal matrices. By using the replica method in
conjunction with the development of an integral formula for handling the random
orthogonal matrices, we show that the concatenated matrices can result in
better recovery performance than what the universality predicts when the
density of non-zero signals is not uniform among the $\nb$ matrix modules. The
universal condition is reproduced for the special case of uniform non-zero
signal densities. Extensive numerical experiments support the theoretical
predictions.
