提出对角加权GMM，高效稳定地处理高维混合高斯模型参数估计。
Diagonally-Weighted Generalized Method of Moments Estimation for Gaussian Mixture Modeling
- 用对角加权替代传统GMM的全矩阵加权，降低计算复杂度。
- 在高维数据下，估计误差更小且运行时间显著缩短。
- 适合需要高效稳健参数估计的高维统计建模场景。
自皮尔逊首次将矩方法（MM）用于一维高斯混合模型以来，基于矩的方法不断涌现。其中广义矩方法（GMM）通过合理加权矩来提升统计效率。然而，MM与GMM的计算复杂度和存储开销随维度增长呈指数级上升，尤其在高维数据或需高阶矩时难以应用。这一问题在GMM中更为严重，因其还需估计大型权重矩阵。为此，本文提出对角加权广义矩方法（DGMM），在统计效率、计算复杂度与数值稳定性间取得平衡。将DGMM应用于弱分离异方差低秩高斯混合模型的参数估计，设计了一种无需显式计算或存储矩张量的高效算法。实验验证表明，相比传统MM与GMM，DGMM在相同条件下实现更小估计误差，且运行时间大幅减少。代码与数据将在发表后公开于https://github.com/liu-lzhang/dgmm。
原文摘要 · Abstract (English)
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.
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