  Supervised dimensionality reduction has emerged as an important theme in the
last decade. Despite the plethora of models and formulations, there is a lack
of a simple model which aims to project the set of patterns into a space
defined by the classes (or categories). To this end, we set up a model in which
each class is represented as a 1D subspace of the vector space formed by the
features. Assuming the set of classes does not exceed the cardinality of the
features, the model results in multi-class supervised learning in which the
features of each class are projected into the class subspace. Class
discrimination is automatically guaranteed via the imposition of orthogonality
of the 1D class sub-spaces. The resulting optimization problem - formulated as
the minimization of a sum of quadratic functions on a Stiefel manifold - while
being non-convex (due to the constraints), nevertheless has a structure for
which we can identify when we have reached a global minimum. After formulating
a version with standard inner products, we extend the formulation to
reproducing kernel Hilbert spaces in a straightforward manner. The optimization
approach also extends in a similar fashion to the kernel version. Results and
comparisons with the multi-class Fisher linear (and kernel) discriminants and
principal component analysis (linear and kernel) showcase the relative merits
of this approach to dimensionality reduction.
