In this paper, we study the problem of learning multi-dimensional Gaussian Mixture Models (GMMs), with a specific focus on model order selection and efficient mixing distribution estimation. We first establish an information-theoretic lower bound on the critical sample complexity required for reliable model selection. More specifically, we show that distinguishing a $k$-component mixture from a simpler model necessitates a sample size scaling of $Ω(Δ^{-(4k-4)})$. We then propose a thresholding-based estimation algorithm that evaluates the spectral gap of an empirical covariance matrix constructed from random Fourier measurement vectors. This parameter-free estimator operates with an efficient time complexity of $\mathcal{O}(k^2 n)$, scaling linearly with the sample size. We demonstrate that the sample complexity of our method matches the established lower bound, confirming its minimax optimality with respect to the component separation distance $Δ$. Conditioned on the estimated model order, we subsequently introduce a gradient-based minimization method for parameter estimation. To effectively navigate the non-convex objective landscape, we employ a data-driven, score-based initialization strategy that guarantees rapid convergence. We prove that this method achieves the optimal parametric convergence rate of $\mathcal{O}_p(n^{-1/2})$ for estimating the component means. To enhance the algorithm's efficiency in high-dimensional regimes where the ambient dimension exceeds the number of mixture components (i.e., \(d > k\)), we integrate principal component analysis (PCA) for dimension reduction. Numerical experiments demonstrate that our Fourier-based algorithmic framework outperforms conventional Expectation-Maximization (EM) methods in both estimation accuracy and computational time.

翻译：本文研究了多维高斯混合模型（GMMs）的学习问题，特别关注模型阶数选择与高效混合分布估计。首先，我们建立了可靠模型选择所需关键样本复杂度的信息论下界。具体地，证明从k分量混合模型区分出更简单模型需要样本量至少为$Ω(Δ^{-(4k-4)})$。随后提出一种基于阈值估计的算法，通过评估随机傅里叶测量向量构建的经验协方差矩阵的谱间隙。该无参数估计器具有$\mathcal{O}(k^2 n)$的高效时间复杂度，与样本量呈线性关系。我们证明该方法的样本复杂度与所建立的下界匹配，证实了其在分量分离距离Δ意义下的极小化最优性。在估计的模型阶数条件下，进一步提出基于梯度的参数估计最小化方法。为有效处理非凸目标函数，采用数据驱动的基于分数的初始化策略，保证快速收敛。证明该方法在估计分量均值时达到最优参数收敛速率$\mathcal{O}_p(n^{-1/2})$。针对高维场景（当环境维度大于混合分量数，即d>k），引入主成分分析（PCA）进行降维以提升算法效率。数值实验表明，基于傅里叶的算法框架在估计精度和计算时间上均优于传统期望最大化（EM）方法。