Structured nonnegative matrix factorization with applications to hidden Markov realization and clustering Original Research Article

In this paper, we study the structured nonnegative matrix factorization problem: given a square, nonnegative matrix P, decompose it as P=VAVinverted perpendicular with V and A nonnegative matrices and with the dimension of A as small as possible. We propose an iterative approach that minimizes the Kullback–Leibler divergence between P and VAVinverted perpendicular subject to the nonnegativity constraints on A and V with the dimension of A given. The approximate structured decomposition Psimilar, equalsVAVinverted perpendicular is closely related to the approximate symmetric decomposition Psimilar, equalsVVinverted perpendicular. It is shown that the approach for finding an approximate structured decomposition can be adapted to solve the symmetric decomposition problem approximately. Finally, we apply the nonnegative decomposition VAVinverted perpendicular to the hidden Markov realization problem and to the clustering of data vectors based on their distance matrix.