  We establish exact asymptotic expressions for the normalized mutual
information and minimum mean-square-error (MMSE) of sparse linear regression in
the sub-linear sparsity regime. Our result is achieved by a generalization of
the adaptive interpolation method in Bayesian inference for linear regimes to
sub-linear ones. A modification of the well-known approximate message passing
algorithm to approach the MMSE fundamental limit is also proposed, and its
state evolution is rigorously analyzed. Our results show that the traditional
linear assumption between the signal dimension and number of observations in
the replica and adaptive interpolation methods is not necessary for sparse
signals. They also show how to modify the existing well-known AMP algorithms
for linear regimes to sub-linear ones.
