This study explores the classification error of Mixture Discriminant Analysis (MDA) in scenarios where the number of mixture components exceeds those present in the actual data distribution, a condition known as overspecification. We use a two-component Gaussian mixture model within each class to fit data generated from a single Gaussian, analyzing both the algorithmic convergence of the Expectation-Maximization (EM) algorithm and the statistical classification error. We demonstrate that, with suitable initialization, the EM algorithm converges exponentially fast to the Bayes risk at the population level. Further, we extend our results to finite samples, showing that the classification error converges to Bayes risk with a rate $n^{-1/2}$ under mild conditions on the initial parameter estimates and sample size. This work provides a rigorous theoretical framework for understanding the performance of overspecified MDA, which is often used empirically in complex data settings, such as image and text classification. To validate our theory, we conduct experiments on remote sensing datasets.

翻译：本研究探讨了混合判别分析在混合成分数量超过实际数据分布中真实成分数量（即过指定条件）下的分类误差。我们在每个类别内使用双组分高斯混合模型来拟合由单一高斯分布生成的数据，同时分析了期望最大化算法的算法收敛性和统计分类误差。我们证明，在适当的初始化条件下，EM算法在总体水平上以指数级速度收敛至贝叶斯风险。进一步，我们将结果推广至有限样本情形，表明在初始参数估计和样本量满足温和条件时，分类误差以$n^{-1/2}$的速率收敛至贝叶斯风险。这项工作为理解过指定MDA的性能提供了严格的理论框架，该框架在图像与文本分类等复杂数据场景中具有广泛经验应用。为验证理论，我们在遥感数据集上进行了实验验证。