  The support vector machine (SVM) is a powerful and widely used classification
algorithm. This paper uses the Karush-Kuhn-Tucker conditions to provide
rigorous mathematical proof for new insights into the behavior of SVM. These
insights provide perhaps unexpected relationships between SVM and two other
linear classifiers: the mean difference and the maximal data piling direction.
For example, we show that in many cases SVM can be viewed as a cropped version
of these classifiers. By carefully exploring these connections we show how SVM
tuning behavior is affected by characteristics including: balanced vs.
unbalanced classes, low vs. high dimension, separable vs. non-separable data.
These results provide further insights into tuning SVM via cross-validation by
explaining observed pathological behavior and motivating improved
cross-validation methodology. Finally, we also provide new results on the
geometry of complete data piling directions in high dimensional space.
