  In this paper, we develop structure assisted nonnegative matrix factorization
(NMF) methods for blind source separation of degenerate data. The motivation
originates from nuclear magnetic resonance (NMR) spectroscopy, where a multiple
mixture NMR spectra are recorded to identify chemical compounds with similar
structures. Consider the linear mixing model (LMM), we aim to identify the
chemical compounds involved when the mixing process is known to be nearly
singular. We first consider a class of data with dominant interval(s) (DI)
where each of source signals has dominant peaks over others. Besides, a nearly
singular mixing process produces degenerate mixtures. The DI condition implies
clustering structures in the data points. Hence, the estimation of the mixing
matrix could be achieved by data clustering. Due to the presence of the noise
and the degeneracy of the data, a small deviation in the estimation may
introduce errors in the output. To resolve this problem and improve robustness
of the separation, methods are developed in two aspects. One is to find better
estimation of the mixing matrix by allowing a constrained perturbation to the
clustering output, and it can be achieved by a quadratic programming. The other
is to seek sparse source signals by exploiting the DI condition, and it solves
an $\ell_1$ optimization. If no source information is available, we propose to
adopt the nonnegative matrix factorization approach by incorporating the matrix
structure (parallel columns of the mixing matrix) into the cost function and
develop multiplicative iteration rules for the numerical solutions. We present
experimental results of NMR data to show the performance and reliability of the
method in the applications arising in NMR spectroscopy.
