Singularity resolution for dimension reduction

Manifold clustering is often used to partition a multiple manifold dataset prior to the application of manifold learning. Thus manifold clustering can be seen as a preprocessing step for eliminating singularities in a dataset before doing dimension reduction. In this paper, we propose an algorithm for resolving singularities prior to dimension reduction. We achieve singularity resolution using algebraic blow ups as motivation. With this type of singularity resolution, we are able to simultaneously perform manifold clustering and learning. The algorithm is based on a simple modification of Isomap which identifies and treats singularities before providing reduced dimensional representations. We demonstrate our algorithm with various examples and apply it to problems in molecular conformation, motion segmentation, and face clustering.