  Decision tree (DT) attracts persistent research attention due to its
impressive empirical performance and interpretability in numerous applications.
However, the growth of traditional yet widely-used univariate decision trees
(UDTs) is quite time-consuming as they need to traverse all the features to
find the splitting value with the maximal reduction of the impurity at each
internal node. In this paper, we newly design a splitting criterion to speed up
the growth. The criterion is induced from Geometric Mean Metric Learning (GMML)
and then optimized under its diagonalized metric matrix constraint,
consequently, a closed-form rank of feature discriminant abilities can at once
be obtained and the top 1 feature at each node used to grow an intent DT
(called as dGMML-DT, where d is an abbreviation for diagonalization). We
evaluated the performance of the proposed methods and their corresponding
ensembles on benchmark datasets. The experiment shows that dGMML-DT achieves
comparable or better classification results more efficiently than the UDTs with
10x average speedup. Furthermore, dGMML-DT can straightforwardly be extended to
its multivariable counterpart (dGMML-MDT) without needing laborious operations.
