  In compressed sensing one measures sparse signals directly in a compressed
form via a linear transform and then reconstructs the original signal. However,
it is often the case that the linear transform itself is known only
approximately, a situation called matrix uncertainty, and that the measurement
process is noisy. Here we present two contributions to this problem: first, we
use the replica method to determine the mean-squared error of the Bayes-optimal
reconstruction of sparse signals under matrix uncertainty. Second, we consider
a robust variant of the approximate message passing algorithm and demonstrate
numerically that in the limit of large systems, this algorithm matches the
optimal performance in a large region of parameters.
