提出高效方法估算高维混合高斯模型的成分均值与数量，适合成分间有明显分离的场景。
Model Selection and Parameter Estimation for Multidimensional Gaussian Mixture Models with a Common Covariance Matrix
- 用傅里叶协方差矩阵识别成分数量，通过谱阈值法估计
- 在成分分离度固定时，均值估计达到最优参数率 $\mathcal{O}_p(n^{-1/2})$
- 比EM算法更高效，适用于多维且成分区分明显的数据
研究具有已知公共协方差矩阵的高维高斯混合模型的模型阶数选择与分量均值估计问题。利用经验特征函数构造傅里叶协方差矩阵，其总体秩等于混合成分数。建立极小极大下界：要区分 $k$ 成分与 $(k-1)$ 成分混合模型，至少需要 $Ω(Δ^{-(4k-4)})$ 个样本。随后提出一个理想谱阈值估计器，当样本量为 $Δ^{-(8k-8)}$ 量级时有效；并设计一种实用的奇异值比估计器。在确定模型阶数后，通过音乐型投影目标函数，结合得分初始化梯度下降估计成分均值。在明确样本量条件下，合格初始点以高概率落在可信吸引域内，迭代线性收敛。对于固定的正成分分离度，均值估计达到参数速率 $\mathcal{O}_p(n^{-1/2})$。数值实验显示，在多种多维设置下，该方法精度优于或相当，计算成本低于期望最大化（EM）算法。
原文摘要 · Abstract (English)
We study model-order selection and component-mean estimation for multidimensional Gaussian mixture models with a known common covariance matrix. Using empirical characteristic-function measurements, we construct Fourier covariance matrices whose population counterparts have rank equal to the number of mixture components. We establish a minimax lower bound showing that distinguishing a separated $k$-component mixture from the class of $(k-1)$-component mixtures requires $Ω(Δ^{-(4k-4)})$ samples. We then develop an oracle spectral-thresholding estimator with a sufficient sample size of order $Δ^{-(8k-8)}$ for fixed $k$, together with a practical singular-value-ratio estimator. Given the model order, we estimate the component means by score-initialized gradient descent on a MUSIC-type projection objective. Under an explicit sample-size condition, a qualifying sample initialization lies in a certified attraction region with high probability, after which the iterates converge linearly. For fixed positive component separation, the resulting mean estimates achieve the parametric rate $\mathcal{O}_p(n^{-1/2})$. Numerical experiments demonstrate competitive accuracy and lower computational cost than expectation-maximization across a range of multidimensional settings.
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