Fundamental limits and algorithms for sparse linear regression with sublinear sparsity
This provides theoretical foundations and practical algorithms for sparse signal recovery in high-dimensional statistics, with applications in compressed sensing and machine learning.
The paper establishes exact asymptotic expressions for mutual information and minimum mean-square-error in sparse linear regression with sublinear sparsity, showing that traditional linear assumptions are unnecessary for sparse signals and proposing a modified AMP algorithm to approach these fundamental limits.
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.