We begin the chapter with some history of the results that form the background of this book. We then define higher matrix factorizations, our main focus. While classical matrix factorizations were factorizations of a single element, higher matrix factorizations deal directly with sequences of elements. In section 1.3, we outline our main results. Throughout the book, we use the notation introdu...

classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as...

classes of‎ ‎abaffy-broyden-spedicato (abs) methods have been introduced for‎ ‎solving linear systems of equations‎. ‎the algorithms are powerful methods for developing matrix‎ ‎factorizations and many fundamental numerical linear algebra processes‎. ‎here‎, ‎we show how to apply the abs algorithms to devise algorithms to compute the wz and zw‎ ‎factorizations of a nonsingular matrix as well as...

Low-rank matrix factorizations such as Principal Component Analysis (PCA), Singular Value Decomposition (SVD) and Non-negative Matrix Factorization (NMF) are a large class of methods for pursuing the low-rank approximation of a given data matrix. The conventional factorization models are based on the assumption that the data matrices are contaminated stochastically by some type of noise. Thus t...

for any given finite abelian group‎, ‎we give factorizations of the group determinant in the group algebra of any subgroups‎. ‎the factorizations is an extension of dedekind's theorem‎. ‎the extension leads to a generalization of dedekind's theorem‎.

in this paper, we extend the construction of a fuzzy subgroup generated by a fuzzy subset to $l$-setting. this construction is illustrated by an example. we also prove that for an $l$-subset of a group, the subgroup generated by its level subset is the level subset of the subgroup generated by that $l$-subset provided the given $l$-subset possesses sup-property.

In this paper we use orderly algorithms to enumerate (perfect) one-factorizations of complete graphs, the automorphism groups of which contain certain prescribed subgroups. We showed that, for the complete graph Ki2, excluding those one-factorizations containing exactly one automorphism of six disjoint cycles of length two, there are precisely 56391 nonisomorphic one-factorizations of Ki2 with ...

Holey factorizations of Kv1,v2,...,vn are a basic building block in the construction of Room frames. In this paper we give some necessary conditions for the existence of holey factorizations and give a complete enumeration for nonisomorphic sets of orthogonal holey factorizations of several special types.

We present a determinization construction for weighted tree automata using factorizations. Among others, this result subsumes a previous result for determinization of weighted string automata using factorizations (Kirsten and Mäurer, 2005) and two previous results for weighted tree automata, one of them not using factorizations (Borchardt, 2004) and one of them restricted to nonrecursive automa...

We present a novel approach to collaborative prediction, using low-norm instead of low-rank factorizations. The approach is inspired by, and has strong connections to, large-margin linear discrimination. We show how to learn low-norm factorizations by solving a semi-definite program, and present generalization error bounds based on analyzing the Rademacher complexity of low-norm factorizations.