The generalized method of moments is (almost) statistically efficient in low-SNR Gaussian latent-variable models

Abstract

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.