This paper addresses the estimation of signals with sublinear sparsity sent over the additive white Gaussian noise channel. This fundamental problem arises in designing denoisers used in message-passing algorithms for sublinear sparsity. The main results are direct and converse theorems in the sublinear sparsity limit, where the signal sparsity grows sublinearly in the signal dimension as the signal dimension tends to infinity. As a direct theorem, the maximum likelihood estimator is proved to achieve vanishing square error in the sublinear sparsity limit if the noise variance is smaller than a threshold. As a converse theorem, all estimators cannot achieve square errors smaller than the signal power under a mild condition if the noise variance is larger than another threshold. In particular, the two thresholds coincide with each other when non-zero signals have constant amplitude. These results imply the asymptotic optimality of an existing separable Bayesian estimator used in approximate message-passing for sublinear sparsity.

翻译：本文研究了通过加性高斯白噪声信道传输的亚线性稀疏信号的估计问题。这一基础问题在设计用于亚线性稀疏度的消息传递算法的去噪器时出现。主要结果是在亚线性稀疏极限下的正定理与逆定理，其中当信号维度趋于无穷时，信号稀疏度随信号维度亚线性增长。作为正定理，本文证明了若噪声方差小于某一阈值，则最大似然估计器在亚线性稀疏极限下可实现趋于零的平方误差。作为逆定理，本文证明了若噪声方差大于另一阈值，则在温和条件下所有估计器均无法获得小于信号功率的平方误差。特别地，当非零信号具有恒定幅度时，这两个阈值彼此重合。这些结果意味着，现有用于亚线性稀疏度近似消息传递的可分离贝叶斯估计器具有渐近最优性。