  In this paper, we propose to study a new geometric optimization problem
called "geometric prototype" in Euclidean space. Given a set of patterns, where
each pattern is represented by a (weighted or unweighted) point set, the
geometric prototype can be viewed as the "mean pattern" minimizing the total
matching cost to them. As a general model, the problem finds many applications
in the areas like machine learning, data mining, computer vision, etc. The
dimensionality could be either constant or high, depending on the applications.
To our best knowledge, the general geometric prototype problem has yet to be
seriously considered by the theory community. To bridge the gap between theory
and practice, we first show that a small core-set can be obtained to
substantially reduce the data size. Consequently, any existing heuristic or
algorithm can run on the core-set to achieve a great improvement on the
efficiency. As a new application of core-set, it needs to tackle a couple of
challenges particularly in theory. Finally, we test our method on both 2D image
and high dimensional clustering datasets; the experimental results remain
stable even if we run the algorithms on the core-sets much smaller than the
original datasets, while the running times are reduced significantly.
