Abstract: We propose a new sufficient dimension reduction approach designed deliberately for high-dimensional classification problems. This novel method is named as Maximal Mean Variance (MMV), inspired by the mean variance index first proposed by Cui, Li and Zhong (2015). MMV requires reasonably mild restrictions on the predictors, and keeps the model-free advantage without the need to estimate the link function. The consistency of the MMV estimator is established under regularity conditions with possibly diverging number of predictors and categories of the response. We also construct the asymptotic normality for the estimator when the dimension of the predictors keeps fixed. The relationship between MMV and several classical classification algorithms are further elaborated. Moreover, although without any definite theoretical guarantee, our method works pretty well when the sample size is far less than the problem dimension. The surprising classification efficiency gain of MMV is demonstrated by simulation studies and real data analysis.

Key words and phrases: Classification, consistency, mean variance index, sufficient dimension reduction.