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 simple 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. 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：我们在亚线性聚变制度中,为正常的相互信息以及稀薄线性回归的最小平均平方-弧度(MMSE)确定精确的空洞表达方式,我们的结果是通过在巴伊西亚线性制度对线性制度的推论中简单归纳适应性内插法到亚线性制度的推论来实现的。还提出了修改众所周知的近似电文传递算法以接近MMSE基本限值的建议。我们的结果显示,复制和适应性内插方法中的信号尺寸和观测次数之间的传统线性假设对于稀释信号是没有必要的。它们还表明如何修改现有的线性制度至亚线性制度的已知AMP算法。