  Discriminative linear models are a popular tool in machine learning. These
can be generally divided into two types: The first is linear classifiers, such
as support vector machines, which are well studied and provide state-of-the-art
results. One shortcoming of these models is that their output (known as the
'margin') is not calibrated, and cannot be translated naturally into a
distribution over the labels. Thus, it is difficult to incorporate such models
as components of larger systems, unlike probabilistic based approaches. The
second type of approach constructs class conditional distributions using a
nonlinearity (e.g. log-linear models), but is occasionally worse in terms of
classification error. We propose a supervised learning method which combines
the best of both approaches. Specifically, our method provides a distribution
over the labels, which is a linear function of the model parameters. As a
consequence, differences between probabilities are linear functions, a property
which most probabilistic models (e.g. log-linear) do not have.
  Our model assumes that classes correspond to linear subspaces (rather than to
half spaces). Using a relaxed projection operator, we construct a measure which
evaluates the degree to which a given vector 'belongs' to a subspace, resulting
in a distribution over labels. Interestingly, this view is closely related to
similar concepts in quantum detection theory. The resulting models can be
trained either to maximize the margin or to optimize average likelihood
measures. The corresponding optimization problems are semidefinite programs
which can be solved efficiently. We illustrate the performance of our algorithm
on real world datasets, and show that it outperforms 2nd order kernel methods.
