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

Fundamental limits and algorithms for sparse linear regression with sublinear sparsity