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

2026-05-28

AI Summary

This work investigates the statistical efficiency of the Generalized Method of Moments (GMoM) for parameter estimation in Gaussian latent variable models—such as Gaussian mixture models and orbit recovery problems—under low signal-to-noise ratio (SNR) conditions, where conventional estimators often perform poorly. The study reveals a hierarchical local geometric structure in the parameter space, wherein distinct moment orders encode information along different directions. By selecting the lowest informative moment order and applying optimal weighting, the proposed GMoM estimator achieves first-order asymptotic covariance matching that of the maximum likelihood estimator. Thus, it attains near-optimal statistical efficiency in low-SNR regimes while preserving the computational advantages inherent to moment-based methods. The theoretical contribution lies in establishing the equivalence between GMoM and maximum likelihood through their shared local expansion of the information operator.

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.

Problem

• low-SNR
• Gaussian latent-variable models
• statistical efficiency
• parameter estimation
• moment-based estimation

Innovation

• generalized method of moments
• low-SNR regime
• statistical efficiency
• layered local geometry
• Gaussian latent-variable models