An infinite matrix is called irreducible if its directed graph is strongly connected. It is proved that an infinite Toeplitz matrix is irreducible if and only if almost every finite leading submatrix is irreducible. An infinite Hankel matrix may be irreducible even if all its finite leading submatrices are reducible. Irreducibility results are also obtained in the finite cases.

We use a multiscale approach to reduce the time to produce the nonnegative matrix factorization (NMF) of a matrix A, that is, A ≈ WH. We also investigate QR factorization as a method for initializing W during the iterative process for producing the nonnegative matrix factorization of A. Finally, we use our approach to produce nonnegative matrix factorizations for classifying images and compare ...

Given a nonnegative matrix factorization, , and factorization rank, exact (exact NMF) decomposes as the product of two matrices, with columns, such . A central research topic in literature is conditions under which decomposition unique/identifiable up to trivial ambiguities. In this paper, we focus on partial identifiability, that is, uniqueness subset columns We start our investigations data‐b...

Given a nonnegative matrix M , the orthogonal nonnegative matrix factorization (ONMF) problem consists in finding a nonnegative matrix U and an orthogonal nonnegative matrix V such that the product UV is as close as possible to M . The importance of ONMF comes from its tight connection with data clustering. In this paper, we propose a new ONMF method, called ONP-MF, and we show that it performs...

The exact nonnegative matrix factorization (exact NMF) problem is the following: given an m-by-n nonnegative matrix X and a factorization rank r, find, if possible, an m-by-r nonnegative matrix W and an r-by-n nonnegative matrix H such that X = WH . In this paper, we propose two heuristics for exact NMF, one inspired from simulated annealing and the other from the greedy randomized adaptive sea...