Analysis on Non-negative Factorizations and Applications

In this work we apply non-negative matrix factorizations (NMF) to some imaging and inverse problems. We propose a sparse low-rank approximation of big data and images in terms of tensor products, and investigate its effectiveness in terms of the number of tensor products to be used in the approximation. A multi-resolution analysis (MRA) framework is presented using a sparse low-rank approximation. We propose a primal-dual active set semi-smooth Newton method for the nonnegative factorization. Numerical results are given to demonstrate the effectiveness of the proposed method to capture features in images and structures of inverse problems under no a-priori assumption on the data structure, as well as to provide a sparse low-rank representation of the data. Mathematics Subject Classification (MSC2000): 15A23, 65F22, 65F30, 65F50, 78M25.