Non-negative matrix factorization considering given vectors in a basis

Recently, a novel matrix factorization, named non-negative matrix factorization (NMF), attracts much attention in the field of signal processing. A matrix with non-negative elements can be decomposed into a product of two matrices with non-negative elements by the NMF. One resulting matrix can be regarded as a basis matrix; and the other can be regarded as a coefficient matrix giving linear combinations of the basis vectors. In practical problems, there exists a case where an ideal basis is partially known. In this paper, we propose a novel method for NMF considering given vectors in an ideal basis. We introduce a criterion for the method and construct an algorithm to optimize the criterion. Moreover, we prove that the proposed algorithm surely converges. Some results of computer simulations are also given to verify the efficacy of the proposed method.