  We present a subspace method to solve large-scale trace ratio problems. This
method is matrix-free, only needing the action of the two matrices in the trace
ratio. At each iteration, a smaller trace ratio problem is addressed in the
search subspace. Additionally, our algorithm is endowed with a restarting
strategy, that ensures the monotonicity of the trace ratio value throughout the
iterations. We also investigate the behavior of the approximate solution from a
theoretical viewpoint, extending existing results on Ritz values and vectors,
as the angle between the search subspace and the exact solution to the trace
ratio approaches zero. In the context of multigroup classification, numerical
experiments show that the new subspace method tends to be more efficient than
iterative approaches that need a (partial) eigenvalue decomposition in every
step.
