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：我们为在亚线性聚变制度中的正常相互信息以及细线性回归的微线性线性回归(MMSE)确定精确的线性表达方式。我们的结果是通过在巴伊西亚对线性制度和亚线性制度的推论中将适应性内插法普遍化来实现的。我们还建议修改众所周知的近似电文传递算法,以接近MMSE基本限度,并严格分析其状态演变。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有信号是不必要的。他们还表明如何修改目前众所周知的线性制度至亚线性制度的AMP算法。