On Clustering and Embedding Manifolds using a Low Rank Neighborhood Approach

In the manifold learning community there has been an onus on the simultaneous clustering and embedding of multiple manifolds. Manifold clustering and embedding algorithms perform especially poorly when embedding highly nonlinear manifolds. In this paper we propose a novel algorithm for improved manifold clustering and embedding. Since a majority of these algorithms are graph based they use different strategies to ensure that only data-point belonging to the same manifold are chosen as neighbors. The new algorithm proposes the addition of a low-rank criterion on the neighborhood of each data-point to ensure that only data-points belonging to the same manifold are "prioritized" for neighbor selection. Following this a reconstruction matrix is calculated to express each data-point as an affine combination of its neighbors. If the low rank neighborhood criterion succeeds in prioritizing data-points belonging to same manifold as neighbors, the reconstruction matrix is (near) block diagonal. This reconstruction matrix can then be used for clustering and embedding. Over a variety of simulated and real data-sets the algorithm shows improvements on the state-of-the-art manifold clustering and embedding algorithms in terms of both clustering and embedding performance.