  In several applications, input samples are more naturally represented in
terms of similarities between each other, rather than in terms of feature
vectors. In these settings, machine-learning algorithms can become very
computationally demanding, as they may require matching the test samples
against a very large set of reference prototypes. To mitigate this issue,
different approaches have been developed to reduce the number of required
reference prototypes. Current reduction approaches select a small subset of
representative prototypes in the space induced by the similarity measure, and
then separately train the classification function on the reduced subset.
However, decoupling these two steps may not allow reducing the number of
prototypes effectively without compromising accuracy. We overcome this
limitation by jointly learning the classification function along with an
optimal set of virtual prototypes, whose number can be either fixed a priori or
optimized according to application-specific criteria. Creating a super-sparse
set of virtual prototypes provides much sparser solutions, drastically reducing
complexity at test time, at the expense of a slightly increased complexity
during training. A much smaller set of prototypes also results in
easier-to-interpret decisions. We empirically show that our approach can reduce
up to ten times the complexity of Support Vector Machines, LASSO and ridge
regression at test time, without almost affecting their classification
accuracy.
