Optimality, Computation, and Interpretation of Nonnegative Matrix Factorizations

The notion of low rank approximations arises from many important applications. When the low rank data are further required to comprise nonnegative values only, the approach by nonnegative matrix factorization is particularly appealing. This paper intends to bring about three points. First, the theoretical Kuhn-Tucker optimality condition is described in explicit form. Secondly, a number of numerical techniques, old and new, are suggested for the nonnegative matrix factorization problems. Thirdly, the techniques are employed to two real-world applications to demonstrate the difficulty in interpreting the factorizations.