Curavture-Aware Non-Negative Matrix Factorization for Clustering

Abstract

Nonnegative matrix factorization (NMF) is one of the most successful techniques in pattern recognition and computer vision. The main advantage of NMF is the parts-based representation of the input, which is consistent with how the human brain recognizes objects. On the other hand, the high dimensional data usually resides on a low dimensional manifold. The performance of the standard NMF can be significantly improved by incorporating with the manifold regularization. However, most existing manifold methods fail to take the extrinsic geometry into account, i.e., how the manifold embedded in the original high dimensional data space, which can discriminative the proximal points from different clusters. In this paper, we propose a novel algorithm, named curvature-aware nonnegative matrix factorization (CANMF), to explicitly consider the extrinsic geometrical structures of the data distribution. First, we build an affinity graph to encode the intrinsic geometrical structure of the data. Then, an anisotropic diffusion process is utilized to exploit the extrinsic curvature, which rescales the weights of the affinity graph by the diffusivity operator. Thus, the weights from different classes can be compressed, and enhance the discriminative ability of the proposed CANMF. The experimental results on several real-world datasets show the advantages of our algorithm.