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.

翻译：我们建立了在次线性稀疏度场景下稀疏线性回归的归一化互信息与最小均方误差的精确渐近表达式。该结果通过将贝叶斯推断中适用于线性场景的自适应插值方法推广至次线性场景而实现。本文还提出了一种对经典近似消息传递算法的改进方案以逼近最小均方误差基本极限，并对其状态演化进行了严格分析。我们的结果表明，在副本法及自适应插值方法中，信号维度与观测数量之间的传统线性假设对于稀疏信号并非必要。这些结果同时揭示了如何将现有适用于线性场景的经典AMP算法改造为适用于次线性场景的版本。