Analysis on a Nonnegative Matrix Factorization and Its Applications

Abstract

In this work we perform some mathematical analysis on a special nonnegative matrix trifactorization (NMF) and apply this NMF to some imaging and inverse problems. We will propose a sparse low-rank approximation of positive data and images in terms of tensor products of positive vectors and investigate its effectiveness in terms of the number of tensor products to be used in the approximation. A new multilevel analysis (MLA) framework is suggested to extract major components in the matrix representing structures of different resolutions but still preserve the positivity of the basis and sparsity of the approximation. We will also propose and formulate a semismooth Newton method based on primal-dual active sets for the nonnegative factorization. Numerical results are given to demonstrate the effectiveness of the proposed method at capturing features in images and structures of inverse problems under no a priori assumption on the underlying structure in the data as well as to provide a sparse low-rank representation of the data.