  Applications of non-linear kernel Support Vector Machines (SVMs) to large
datasets is seriously hampered by its excessive training time. We propose a
modification, called the approximate extreme points support vector machine
(AESVM), that is aimed at overcoming this burden. Our approach relies on
conducting the SVM optimization over a carefully selected subset, called the
representative set, of the training dataset. We present analytical results that
indicate the similarity of AESVM and SVM solutions. A linear time algorithm
based on convex hulls and extreme points is used to compute the representative
set in kernel space. Extensive computational experiments on nine datasets
compared AESVM to LIBSVM \citep{LIBSVM}, CVM \citep{Tsang05}, BVM
\citep{Tsang07}, LASVM \citep{Bordes05}, $\text{SVM}^{\text{perf}}$
\citep{Joachims09}, and the random features method \citep{rahimi07}. Our AESVM
implementation was found to train much faster than the other methods, while its
classification accuracy was similar to that of LIBSVM in all cases. In
particular, for a seizure detection dataset, AESVM training was almost $10^3$
times faster than LIBSVM and LASVM and more than forty times faster than CVM
and BVM. Additionally, AESVM also gave competitively fast classification times.
