摘要：In this paper, we propose to study a new geometric optimization problem called the "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 "average pattern" minimizing the total matching cost to them. As a general model, the problem finds many applications in real-world, such as Wasserstein barycenter and ensemble clustering. 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 image and high dimensional clustering datasets; the experimental results remain stable even if we run the algorithms on core-sets much smaller than the original datasets, while the running times are reduced significantly.
