标题： 对角加权广义矩方法在高斯混合建模中的估计 显示英文标题

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

摘要： 自皮尔逊[《皇家学会哲学汇刊》A, 185 (1894), 第71-110页]首次将矩方法(MM)应用于将数据建模为一维高斯混合以来，基于矩的估计方法已经大量出现。在这些方法中，广义矩方法(GMM)通过适当加权矩来提高MM的统计效率。然而，MM和GMM的计算复杂度和存储复杂度随着维度的增加呈指数增长，这使得这些方法在高维数据或需要更高阶矩的情况下变得不切实际。这种计算瓶颈在GMM中更为严重，因为它还需要估计一个大的权重矩阵。为了克服这些瓶颈，我们提出了对角加权GMM(DGMM)，它在统计效率、计算复杂度和数值稳定性之间取得了平衡。我们将DGMM应用于研究弱分离异方差低秩高斯混合的参数估计问题，并设计了一种计算高效且数值稳定的算法，该算法无需显式计算或存储矩张量即可获得DGMM估计量。我们实现了所提出的算法，并通过实验验证了DGMM的优势：在数值研究中，DGMM在要求显著更短运行时间的同时获得了更小的估计误差。代码和数据将在发表后在https://github.com/liu-lzhang/dgmm上提供。 显示英文摘要

摘要： 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.