Multi-group classification arises in many prediction and decision-making problems, including applications in epidemiology, genomics, finance, and image recognition. Although classification methods have advanced considerably, much of the literature focuses on binary problems, and available extensions often provide limited flexibility for multi-group settings. Recent work has extended linear discriminant analysis to multiple groups, but more general methods are still needed to handle complex structures such as nonlinear decision boundaries and group-specific covariance patterns. We develop Multi-Group Quadratic Discriminant Analysis (MGQDA), a method for multi-group classification built on quadratic discriminant analysis. MGQDA projects high-dimensional predictors onto a lower-dimensional subspace, which enables accurate classification while capturing nonlinearity and heterogeneity in group-specific covariance structures. We derive theoretical guarantees, including variable selection consistency, to support the reliability of the procedure. In simulations and a gene-expression application, MGQDA achieves competitive or improved predictive performance compared with existing methods while selecting group-specific informative variables, indicating its practical value for high-dimensional multi-group classification problems. Supplementary materials for this article are available online.

翻译：多组分类问题广泛存在于预测与决策领域，包括流行病学、基因组学、金融和图像识别等应用场景。尽管分类方法已取得显著进展，现有文献多聚焦于二分类问题，而可用的扩展方法往往对多组场景的灵活性有限。近期研究已将线性判别分析扩展至多组情形，但仍需更通用的方法来处理复杂结构，例如非线性决策边界和组间特异性协方差模式。本文提出了多组二次判别分析（MGQDA），这是一种基于二次判别分析构建的多组分类方法。MGQDA将高维预测变量投影至低维子空间，在捕捉组间特异性协方差结构的非线性与异质性的同时实现精确分类。我们推导了包括变量选择一致性在内的理论保证，以支撑该方法的可靠性。在模拟实验和基因表达数据应用中，MGQDA在筛选组间特异性信息变量的同时，相比现有方法取得了具有竞争力或更优的预测性能，表明其在高维多组分类问题中具有实用价值。本文的补充材料可在线获取。