  Dimensionality reduction (DR) is often used as a preprocessing step in
classification, but usually one first fixes the DR mapping, possibly using
label information, and then learns a classifier (a filter approach). Best
performance would be obtained by optimizing the classification error jointly
over DR mapping and classifier (a wrapper approach), but this is a difficult
nonconvex problem, particularly with nonlinear DR. Using the method of
auxiliary coordinates, we give a simple, efficient algorithm to train a
combination of nonlinear DR and a classifier, and apply it to a RBF mapping
with a linear SVM. This alternates steps where we train the RBF mapping and a
linear SVM as usual regression and classification, respectively, with a
closed-form step that coordinates both. The resulting nonlinear low-dimensional
classifier achieves classification errors competitive with the state-of-the-art
but is fast at training and testing, and allows the user to trade off runtime
for classification accuracy easily. We then study the role of nonlinear DR in
linear classification, and the interplay between the DR mapping, the number of
latent dimensions and the number of classes. When trained jointly, the DR
mapping takes an extreme role in eliminating variation: it tends to collapse
classes in latent space, erasing all manifold structure, and lay out class
centroids so they are linearly separable with maximum margin.
