Extension of ISOMAP for Imperfect Manifolds

As one of the most promising nonlinear dimensionality reduction techniques, Isometric Mapping (ISOMAP) performs well only when the data belong to a single well-sampled manifold, where geodesic distances can be well approximated by the corresponding shortest path distances in a suitable neighborhood graph. Unfortunately, the approximation gets less and less precise generally as the number of edges of the corresponding shortest path increases, which makes ISOMAP tend to overlap or overcluster the data, especially for disjoint or imperfect manifolds. To alleviate this problem, this paper presented a variant of ISOMAP, i.e. Edge Number-based ISOMAP (ENISOMAP), which uses a new variant of Multidimensional Scaling (MDS), i.e. Edge Number-based Multidimensional Scaling (EN-MDS), instead of the classical Multidimensional Scaling (CMDS) to map the data into the low-dimensional embedding space. As a nonlinear variant of MDS, ENMDS gives larger weight to the distances with fewer edges, which are generally better approximated and then more trustworthy than those with more edges, and thus can preserve the more trustworthy distances more precisely. Finally, experimental results verify that not only imperfect manifolds but also intrinsically curved manifold can be visualized by EN-ISOMAP well.