In Nonnegative Matrix Factorization (NMF), a nonnegative matrix is approximated by a product of lower-rank factorizing matrices. Most NMF methods assume that each factorizing matrix appears only once in the approximation, thus the approximation is linear in the factorizing matrices. We present a new class of approximative NMF methods, called Quadratic Nonnegative Matrix Factorization (QNMF), wh...

Nonnegative matrix factorization (NMF) is a powerful tool for data mining. However, the emergence of ‘big data’ has severely challenged our ability to compute this fundamental decomposition using deterministic algorithms. This paper presents a randomized hierarchical alternating least squares (HALS) algorithm to compute the NMF. By deriving a smaller matrix from the nonnegative input data, a mo...

The workshop brought together young and experienced researchers who study nonnegative matrix theory and its applications. The speakers at the workshop presented recent progress, open problems and challenges involving nonnegative matrices and their generalizations. Specifically, discussed were eventually nonnegative matrices; combinatorial aspects of nonnegative matrix theory and its interplay w...

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...

Besides Karatsuba algorithm, optimal Toeplitz matrix-vector product (TMVP) formulae is another approach to design GF (2) subquadratic multipliers. However, when GF (2) elements are represented using a shifted polynomial basis, this approach is currently appliable only to GF (2)s generated by all irreducible trinomials and a special type of irreducible pentanomials, not all general irreducible p...

The nonnegative rank of a nonnegative matrix is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively. Such decompositions are useful in diverse scientific disciplines. We obtain characterizations and bounds and show that the nonnegative rank can be computed exactly over the reals by a finite algorithm.

Nonnegative nilpotent lower triangular completions of a nonnegative nilpotent matrix are studied. It is shown that for every natural number between the index of the matrix and its order, there exists a completion that has this number as its index. A similar result is obtained for the rank. However, unlike the case of complex completions of complex matrices, it is proved that for every nonincrea...