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. From a theoretical perspective, 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. This threshold is known to be achievable by an existing separable Bayesian estimator. 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 the existing separable Bayesian estimator used in approximate message-passing for sublinear sparsity. From a numerical perspective, a non-separable estimator is proposed via a heuristic approximation of the true posterior mean estimator. Numerical simulations show that the ML estimator and the proposed non-separable estimator outperform the separable Bayesian estimator for high signal-to-noise ratio (SNR). In the low SNR regime, on the other hand, the two estimators are inferior to the separable Bayesian estimator while the proposed non-separable estimator slightly outperforms the ML estimator.

翻译：本文研究通过加性高斯白噪声信道传输的次线性稀疏信号的估计问题。这一基本问题出现在为次线性稀疏性设计的消息传递算法中的去噪器构建中。从理论视角看，主要结果是在次线性稀疏极限下的正定理与逆定理——当信号维度趋于无穷时，信号稀疏度随维度呈次线性增长。作为正定理，若噪声方差小于某一阈值，最大似然估计器被证明能在次线性稀疏极限下实现平方误差趋于零。该阈值已知可由现有可分离贝叶斯估计器达到。作为逆定理，在温和条件下，若噪声方差大于另一阈值，所有估计器均无法实现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，这两个阈值恰好一致。这些结果揭示了用于次线性稀疏性的近似消息传递中现有可分离贝叶斯估计器的渐近最优性。从数值视角看，通过真实后验均值估计器的启发式近似，本文提出了一种不可分离估计器。数值仿真表明，在高信噪比（SNR）场景下，最大似然估计器与所提不可分离估计器均优于可分离贝叶斯估计器。然而在低信噪比区域，这两种估计器逊于可分离贝叶斯估计器，但所提不可分离估计器略优于最大似然估计器。