  We propose a new sufficient dimension reduction approach designed
deliberately for high-dimensional classification. This novel method is named
maximal mean variance (MMV), inspired by the mean variance index first proposed
by Cui, Li and Zhong (2015), which measures the dependence between a
categorical random variable with multiple classes and a continuous random
variable. Our method requires reasonably mild restrictions on the predicting
variables 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 for both fixed and diverging dimension (p) cases and the
number of the response classes can also be allowed to diverge with the sample
size n. We also construct the asymptotic normality for the estimator when the
dimension of the predicting vector is fixed. Furthermore, our method works
pretty well when n < p. The surprising classification efficiency gain of the
proposed method is demonstrated by simulation studies and real data analysis.
