  Feature selection involes identifying the most relevant subset of input
features, with a view to improving generalization of predictive models by
reducing overfitting. Directly searching for the most relevant combination of
attributes is NP-hard. Variable selection is of critical importance in many
applications, such as micro-array data analysis, where selecting a small number
of discriminative features is crucial to developing useful models of disease
mechanisms, as well as for prioritizing targets for drug discovery. The
recently proposed Minimal Complexity Machine (MCM) provides a way to learn a
hyperplane classifier by minimizing an exact (\boldmath{$\Theta$}) bound on its
VC dimension. It is well known that a lower VC dimension contributes to good
generalization. For a linear hyperplane classifier in the input space, the VC
dimension is upper bounded by the number of features; hence, a linear
classifier with a small VC dimension is parsimonious in the set of features it
employs. In this paper, we use the linear MCM to learn a classifier in which a
large number of weights are zero; features with non-zero weights are the ones
that are chosen. Selected features are used to learn a kernel SVM classifier.
On a number of benchmark datasets, the features chosen by the linear MCM yield
comparable or better test set accuracy than when methods such as ReliefF and
FCBF are used for the task. The linear MCM typically chooses one-tenth the
number of attributes chosen by the other methods; on some very high dimensional
datasets, the MCM chooses about $0.6\%$ of the features; in comparison, ReliefF
and FCBF choose 70 to 140 times more features, thus demonstrating that
minimizing the VC dimension may provide a new, and very effective route for
feature selection and for learning sparse representations.
