We propose a new variant of nonnegative matrix factorization (NMF), combining separability and sparsity assumptions. Separability requires that the columns first NMF factor are equal to input matrix, while second sparse. call this sparse separable (SSNMF), which we prove be NP-complete, as opposed can solved in polynomial time. The main motivation consider model is handle underdetermined blind ...

Nonnegative matrix factorization (NMF) is a linear dimensionality technique for nonnegative data with applications such as image analysis, text mining, audio source separation, and hyperspectral unmixing. Given MM rank rr, NMF looks WW rr columns HH rows that M ≈ WHM≈WH. NP-hard to solve in general. However, it can be computed efficiently under the separability assumption which requires basis v...

Generalizing the idea of viewing a digraph as a model of a linear map, we suggest a multi-variable analogue of a digraph, called a hydra, as a model of a multi-linear map. Walks in digraphs correspond to usual matrix multiplication while walks in hydras correspond to the tensor multiplication introduced by Robert Grone in 1987. By viewing matrix multiplication as a special case of this tensor m...

Let Amx„, Bmxn, Xnxl, and Ymxl be matrices whose entries are nonnegative real numbers and suppose that no row of A and no column of B consists entirely of zeroes. Define the operators U, T and T by (UX)t-X? [or (UY),= Y;1], T=UB'UA and T' = UAUB'. Tis called irreducible if for no nonempty proper subset S of (1, • ■ ■ , n} it is true that X,=0, ieS; X,^0, i$ S, implies (TX),=0, ieS; (TX)i^O, i $...

The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified pr...