Image classification lets us automatically understand what's in an image, but by itself this isn't enough to enable search. Sure, if a user searches for beach we could return the images with the highest scores for that category, but what if they instead search for shore? What if instead of apple they search for fruit or granny smith? We could collate a large dictionary of synonyms and near-synonyms and hierarchical relationships between words, but this quickly becomes unwieldy, especially if we support multiple languages. Word vectors So let's reframe the problem.

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.