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：考虑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颖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，所提方法的优势在于：当类别数较少（特别是二元响应时）能够估计多个有效降维方向。此外，所提方法无需满足苛刻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遇到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。在温和条件下建立了所提方法的统计相合性。通过模拟和实际数据分析表明，与现有的一些函数型充分降维方法相比，这两种方法具有竞争力。