  In this work we perform some mathematical analysis on non-negative matrix
factorizations (NMF) and apply NMF to some imaging and inverse problems. We
will propose a sparse low-rank approximation of big 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 concept of multi-level analysis (MLA) framework is also suggested to
extract major components in the matrix representing structures of different
resolutions, but still preserving the positivity of the basis and sparsity of
the approximation. We will also propose a semi-smooth Newton method based on
primal-dual active sets for the non-negative 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.
