Fundamental limits and algorithms for sparse linear regression with sublinear sparsity

TL;DR

This paper derives exact asymptotic formulas for mutual information and MMSE in sparse linear regression with sublinear sparsity, and proposes algorithmic modifications with rigorous analysis.

Contribution

It generalizes the adaptive interpolation method to sub-linear sparsity regimes and modifies AMP algorithms accordingly.

Findings

Exact asymptotic expressions for mutual information and MMSE.

Modified AMP algorithms approach the MMSE limit in sub-linear regimes.

Linear assumptions are not necessary for sparse signals in these regimes.

Abstract

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.