Considering the case where the response variable is a categorical variable and the predictor is a random function, two novel functional sufficient dimensional reduction (FSDR) methods are proposed based on mutual information and square loss mutual information. Compared to the classical FSDR methods, such as functional sliced inverse regression and functional sliced average variance estimation, the proposed methods are appealing because they are capable of estimating multiple effective dimension reduction directions in the case of a relatively small number of categories, especially for the binary response. Moreover, the proposed methods do not require the restrictive linear conditional mean assumption and the constant covariance assumption. They avoid the inverse problem of the covariance operator which is often encountered in the functional sufficient dimension reduction. The functional principal component analysis with truncation be used as a regularization mechanism. Under some mild conditions, the statistical consistency of the proposed methods is established. It is demonstrated that the two methods are competitive compared with some existing FSDR methods by simulations and real data analyses.

本文提出了两种基于互信息和平方损失互信息的新型功能性充分降维（FSDR）方法，能够在分类数目相对较小的情况下估计多个有效的降维方向，尤其是对于二元响应，而且不需要线性条件平均值和常数协方差假设。在一些温和的条件下，本文建立了所提出方法的统计一致性，并通过模拟和实际数据分析证明了这两种方法的竞争性。

通过信息最大化进行功能充分降维并应用于分类