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：本文研究了加性白高斯噪声信道中亚线性稀疏信号的估计问题。这一基础性问题源于用于亚线性稀疏性的消息传递算法中降噪器的设计。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，信号稀疏度相对于信号维度呈亚线性增长。作为正定理，当噪声方差小于某阈值时，最大似然估计器被证明能在亚线性稀疏极限下实现均方误差趋于零。作为逆定理，当噪声方差大于另一阈值时，在温和条件下所有估计器都无法实现小于信号功率的均方误差。特别地，当非零信号具有恒定幅度时，这两个阈值相互重合。这些结果揭示了现有用于亚线性稀疏性的近似消息传递算法中可分离贝叶斯估计器的渐近最优性。