  Classification is an important topic in statistics and machine learning with
great potential in many real applications. In this paper, we investigate two
popular large margin classification methods, Support Vector Machine (SVM) and
Distance Weighted Discrimination (DWD), under two contexts: the
high-dimensional, low-sample size data and the imbalanced data. A unified
family of classification machines, the FLexible Assortment MachinE (FLAME) is
proposed, within which DWD and SVM are special cases. The FLAME family helps to
identify the similarities and differences between SVM and DWD. It is well known
that many classifiers overfit the data in the high-dimensional setting; and
others are sensitive to the imbalanced data, that is, the class with a larger
sample size overly influences the classifier and pushes the decision boundary
towards the minority class. SVM is resistant to the imbalanced data issue, but
it overfits high-dimensional data sets by showing the undesired data-piling
phenomena. The DWD method was proposed to improve SVM in the high-dimensional
setting, but its decision boundary is sensitive to the imbalanced ratio of
sample sizes. Our FLAME family helps to understand an intrinsic connection
between SVM and DWD, and improves both methods by providing a better trade-off
between sensitivity to the imbalanced data and overfitting the high-dimensional
data. Several asymptotic properties of the FLAME classifiers are studied.
Simulations and real data applications are investigated to illustrate the
usefulness of the FLAME classifiers.
