On the Set of Solutions of the Nonnegative Matrix Factorization Problem.

The nonnegative matrix factorization (NMF) problem D = XYT for a given nonnegative matrix D and with nonnegative factors X and Y can have many solutions aside from trivial permutations or positive multiples of the columns of X and Y. The set of feasible solutions (SFS) is a low-dimensional representation of all possible columns of either X or Y in any NMF of D. The SFS provides important information on the possible ambiguity of the NMF. This paper conveys the SFS concept as developed in chemometrics to mathematics. Various properties of the SFS are proved. Numerical algorithms for the SFS computation are reviewed and tested for an application model problem from analytical chemistry.