Adaptive Kernel Graph Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is an efficient method for feature learning in the field of machine learning and data mining. To investigate the nonlinear characteristics of datasets, kernel-method-based NMF (KNMF) and its graph-regularized extensions have received much attention from various researchers due to their promising performance. However, the graph similarity matrix of the existing methods is often predefined in the original space of data and kept unchanged during the matrix-factorization procedure, which leads to non-optimal graphs. To address these problems, we propose a kernel-graph-learning-based, nonlinear, nonnegative matrix-factorization method in this paper, termed adaptive kernel graph nonnegative matrix factorization (AKGNMF). In order to automatically capture the manifold structure of the data on the nonlinear feature space, AKGNMF learned an adaptive similarity graph. We formulated a unified objective function, in which global similarity graph learning is optimized jointly with the matrix decomposition process. A local graph Laplacian is further imposed on the learned feature subspace representation. The proposed method relies on both the factorization that respects geometric structure and the mapped high-dimensional subspace feature representations. In addition, an efficient iterative solution was derived to update all variables in the resultant objective problem in turn. Experiments on the synthetic dataset visually demonstrate the ability of AKGNMF to separate the nonlinear dataset with high clustering accuracy. Experiments on real-world datasets verified the effectiveness of AKGNMF in three aspects, including clustering performance, parameter sensitivity and convergence. Comprehensive experimental findings indicate that, compared with various classic methods and the state-of-the-art methods, the proposed AKGNMF algorithm demonstrated effectiveness and superiority.