Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling

2025-07-27

AI Summary

Traditional generalized method of moments (GMM) suffers from exponential computational and storage complexity in high-dimensional settings due to explicit construction and manipulation of high-order moment tensors, rendering it impractical. To address this, we propose Diagonal-weighted GMM (DGMM), tailored for parameter estimation in weakly separated, heteroskedastic, low-rank Gaussian mixture models. DGMM replaces the full covariance-based weighting matrix with a learnable diagonal weighting matrix, thereby avoiding explicit formation and storage of high-order moment tensors while preserving statistical efficiency. Leveraging low-rank structural priors and a numerically stable iterative optimization algorithm, DGMM enables scalable and robust estimation. Experiments demonstrate that DGMM achieves significantly lower estimation error than standard GMM and classical moment-based methods, while reducing runtime by one to two orders of magnitude—achieving both high statistical accuracy and computational feasibility.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Reasoning under Uncertainty: Graphical Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Since Pearson [Philosophical Transactions of the Royal Society of London. A, 185 (1894), pp. 71-110] first applied the method of moments (MM) for modeling data as a mixture of one-dimensional Gaussians, moment-based estimation methods have proliferated. Among these methods, the generalized method of moments (GMM) improves the statistical efficiency of MM by weighting the moments appropriately. However, the computational complexity and storage complexity of MM and GMM grow exponentially with the dimension, making these methods impractical for high-dimensional data or when higher-order moments are required. Such computational bottlenecks are more severe in GMM since it additionally requires estimating a large weighting matrix. To overcome these bottlenecks, we propose the diagonally-weighted GMM (DGMM), which achieves a balance among statistical efficiency, computational complexity, and numerical stability. We apply DGMM to study the parameter estimation problem for weakly separated heteroscedastic low-rank Gaussian mixtures and design a computationally efficient and numerically stable algorithm that obtains the DGMM estimator without explicitly computing or storing the moment tensors. We implement the proposed algorithm and empirically validate the advantages of DGMM: in numerical studies, DGMM attains smaller estimation errors while requiring substantially shorter runtime than MM and GMM. The code and data will be available upon publication at https://github.com/liu-lzhang/dgmm.

Problem

• Overcoming computational complexity in high-dimensional Gaussian mixture modeling
• Improving statistical efficiency without large weighting matrices
• Enhancing numerical stability in moment-based estimation methods

Innovation

• Diagonally-weighted GMM balances efficiency and complexity
• Avoids explicit computation of moment tensors
• Faster runtime with smaller estimation errors