在回归模型中考虑解释变量之间的交互效应，是对当前线性模型的自然延伸，很多实际数据也表明解释变量之间的交互效应的确能给应变量带来影响。交互效应模型在经典统计分析中有所研究，例如二次回归分析和二次判别分析等。由于交互项数量随着数据维度呈平方形式增长以及自身具有结构性，高维交互效应模型给当前统计分析带来很大的挑战。本项目拟从回归函数的海森矩阵出发，研究高维情形下交互效应模型的充分性降维问题。特别的对二次回归分析和二次判别分析，海森矩阵对应着模型中的交互效应回归系数矩阵。通过对海森矩阵的研究，建立回归系数矩阵和一类加权协方差矩阵之间的关系。从稀疏和低秩两种不同模型结构下构造出高维二次回归分析和二次判别模型的相合估计以及相关的变量选择方法。基于矩阵的角度来研究高维交互效应模型，所得结果将具有更好的结构性和可解释性，新的方法也将被应用到信息技术、生物医学等领域解决一些实际问题。

In the regression model, considering the interactions between the covariates is a natural extension of the linear model. Many real examples show that the interaction between the explanatory variables can help improve the model interpretability and the prediction. In the statistical analysis, there are many classical models involving the interaction, such as quadratic regression and quadratic discriminant analysis. Since the number of the interactions increase quadratically with the data dimension and the interactions have complex covariance structures, it is a challenging problem to deal with interactions for high dimensional data. In this project, we propose to study the the Hessian matrix of the regression function to conduct a sufficient dimension reduction for high dimensional regression model with interactions. The Hessian matrix describes the second order partial derivatives of the regression function and specially for the quadratic regression and the quadratic discriminant analysis, it relates to the regression coefficient matrix with the interactions. By studying the Hessian matrix, we derive the relationship between the regression coefficient matrix and a class of weighted covariance matrices. Under the sparse or low rank structure, we conduct consistent estimation and variable selection for high dimensional quadratic regression and high dimensional quadratic discriminant analysis. From the perspective of matrix to study high dimensional regressions with interactions, the resulting model will have good structures and model interpretability. The new methods will also be evaluated by the real applications in biomedical and other related areas.