Manifold Peaks Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) has attracted increasing interest for its high interpretability in recent years. It is shown that the NMF is closely related to fuzzy k-means clustering, where the basis matrix represents the cluster centroids. However, most of the existing NMF-based clustering algorithms often have their decomposed centroids deviate away from the data manifold, which potentially undermines the clustering results, especially when the datasets lie on complicated geometric structures. In this article, we present a manifold peaks NMF (MPNMF) for data clustering. The proposed approach has the following advantages: 1) it selects a number of MPs to characterize the backbone of the data manifold; 2) it enforces the centroids to lie on the original data manifold, by restricting each centroid to be a conic combination of a small number of nearby MPs; 3) it generalizes the graph smoothness regularization to guide the local graph construction; and 4) it solves a general problem of quadratic regularized nonnegative least squares (NNLSs) with group l(0)-norm constraint and further develops an efficient optimization algorithm to solve the objective function of the MPNMF. Extensive experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed approach.