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Considering the case where the response variable is a categorical variableand the predictor is a random function, two novel functional sufficientdimensional reduction (FSDR) methods are proposed based on mutual informationand square loss mutual information. Compared to the classical FSDR methods,such as functional sliced inverse regression and functional sliced averagevariance estimation, the proposed methods are appealing because they arecapable of estimating multiple effective dimension reduction directions in thecase of a relatively small number of categories, especially for the binaryresponse. Moreover, the proposed methods do not require the restrictive linearconditional mean assumption and the constant covariance assumption. They avoidthe inverse problem of the covariance operator which is often encountered inthe functional sufficient dimension reduction. The functional principalcomponent analysis with truncation be used as a regularization mechanism. Undersome mild conditions, the statistical consistency of the proposed methods isestablished. It is demonstrated that the two methods are competitive comparedwith some existing FSDR methods by simulations and real data analyses.

Functional sufficient dimension reduction through information maximization with application to classification