  We propose a fast non-gradient-based method of rank-1 non-negative matrix
factorization (NMF) for missing data, called A1GM, that minimizes the KL
divergence from an input matrix to the reconstructed rank-1 matrix. Our method
is based on our new finding of an analytical closed-formula of the best rank-1
non-negative multiple matrix factorization (NMMF), a variety of NMF. NMMF is
known to exactly solve NMF for missing data if positions of missing values
satisfy a certain condition, and A1GM transforms a given matrix so that the
analytical solution to NMMF can be applied. We empirically show that A1GM is
more efficient than a gradient method with competitive reconstruction errors.
