  This paper introduces an algorithm for the nonnegative matrix
factorization-and-completion problem, which aims to find nonnegative low-rank
matrices X and Y so that the product XY approximates a nonnegative data matrix
M whose elements are partially known (to a certain accuracy). This problem
aggregates two existing problems: (i) nonnegative matrix factorization where
all entries of M are given, and (ii) low-rank matrix completion where
nonnegativity is not required. By taking the advantages of both nonnegativity
and low-rankness, one can generally obtain superior results than those of just
using one of the two properties. We propose to solve the non-convex constrained
least-squares problem using an algorithm based on the classic alternating
direction augmented Lagrangian method. Preliminary convergence properties of
the algorithm and numerical simulation results are presented. Compared to a
recent algorithm for nonnegative matrix factorization, the proposed algorithm
produces factorizations of similar quality using only about half of the matrix
entries. On tasks of recovering incomplete grayscale and hyperspectral images,
the proposed algorithm yields overall better qualities than those produced by
two recent matrix-completion algorithms that do not exploit nonnegativity.
