We study estimation in the low signal-to-noise ratio (SNR) regime for a broad class of Gaussian latent-variable models, including Gaussian mixtures and orbit recovery problems. We show that, in this regime, the generalized method-of-moments (GMoM) matches the first-order asymptotic efficiency of maximum likelihood. In particular, if the moment features are chosen up to the minimal local order required for identification and are weighted optimally, then the resulting GMoM estimator has the same leading asymptotic covariance as the maximum-likelihood estimator. Our analysis shows that, in low SNR, this equivalence is governed by a layered local geometry: different directions become informative at different moment orders, partitioning the space into layers with distinct SNR scalings. We prove that the observed Fisher information and the GMoM information operator admit matching layerwise expansions across these layers. As a consequence, in the low-SNR regime, GMoM provides a statistically efficient alternative to maximum likelihood, while preserving the computational advantages of moment-based estimation.

翻译：我们研究了一类广泛的高斯潜变量模型（包括高斯混合模型和轨道恢复问题）在低信噪比（SNR）条件下的估计问题。研究表明，在此条件下，广义矩估计（GMoM）能够达到与最大似然估计相同的一阶渐近效率。特别地，若矩特征选择至可辨识所需的最小局部阶数并经过最优加权，则所得GMoM估计量与最大似然估计量具有相同的主导渐近协方差结构。我们的分析表明，在低信噪比环境下，这种等价性由分层局部几何结构决定：不同方向在不同矩阶数下呈现信息性，将参数空间划分为具有不同信噪比缩放因子的层。我们证明观测Fisher信息矩阵与GMoM信息算子在这些层上具有匹配的分层展开形式。因此，在低信噪比条件下，GMoM在保留矩估计计算优势的同时，提供了统计上有效的最大似然估计替代方案。