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.

翻译：针对响应变量为分类变量、预测变量为随机函数的情形，基于互信息和平方损失互信息提出了两种新的函数型充分降维方法。与经典的函数型充分降维方法（如函数型切片逆回归和函数型切片平均方差估计）相比，所提方法在类别数较少（尤其针对二元响应）的情况下能够估计多个有效降维方向，因而更具优势。此外，所提方法无需满足严格的线性条件均值假设和常数协方差假设，避免了函数型充分降维中常见的协方差算子逆问题。采用截断函数型主成分分析作为正则化机制。在温和条件下，证明了所提方法的统计相合性。通过模拟实验和实际数据分析，证实了这两种方法与现有函数型充分降维方法相比具有竞争力。