Selection of the Suitable Neighborhood Size for the ISOMAP Algorithm

The success of ISOMAP depends greatly on selecting a suitable neighborhood size; however, it´s an open problem how to do this efficiently. When the neighborhood size is unsuitable, shortcut edges can emerge in the neighborhood graph and shorten the involved shortest path lengths greatly, which makes them not approximate the corresponding geodesic distances anymore, that is, there doesn´t exist such an approximately monotonically increasing relationship between them anymore. Based on this observation, in the paper, we use costs over the minimal connected neighborhood graph to approximate the corresponding geodesic distances, and then present an efficient method to judge whether a neighborhood size is suitable beforehand, by which a suitable neighborhood size can be selected more efficiently than the straightforward method with the residual variance. Besides, the correctness of the intrinsic dimensionality, estimated by ISOMAP, of the data can also be judged more easily by our method.