Low-rank Nonnegative Matrix Factorization on Stiefel Manifold

Abstract Low rank is an important but ill-posed problem in the development of nonnegative matrix factorization (NMF) algorithms because the essential information is often encoded in a low-rank intrinsic data matrix, whereas noise and outliers are contained in a residue matrix. Most existing NMF approaches achieve low rank by directly specifying the dimensions of the factor matrices. A few others impose the low rank constraint on the factor matrix and use the alternating direction method of multipliers to solve the optimization problem. In contrast to previous approaches, this paper proposes a novel method for low-rank nonnegative matrix factorization on a Stiefel manifold (LNMFS), which utilizes the low rank structure of intrinsic data and transforms it into a Frobenius norm of the latent factors. To obtain orthogonal factors as distinct patterns, we further impose orthogonality constraints by assuming that the basis matrix lies on a Stiefel manifold. In addition, to improve the robustness of the data in a manifold structure, we incorporate the graph smoothness of the coefficient matrix. Finally, we develop an efficient alternative iterative algorithm to solve the optimization problem and provide proof of its convergence. Extensive experiments on real-world datasets demonstrate the superiority of the proposed method compared with other representative NMF-based algorithms.