Related papers: Elimination and Factorization

The paper introduces the butterfly factorization as a data-sparse approximation for the matrices that satisfy a complementary low-rank property. The factorization can be constructed efficiently if either fast algorithms for applying the…

Factorization of an $n\times n$ unitary matrix as a product of $n$ diagonal matrices containing only phases interlaced with $n-1$ orthogonal matrices each one generated by a real vector as well as an explicit form for the Weyl factorization…

The low-rank matrix factorization as a L1 norm minimization problem has recently attracted much attention due to its intrinsic robustness to the presence of outliers and missing data. In this paper, we propose a new method, called the…

Factoring a matrix into two low rank matrices is at the heart of many problems. The problem of matrix completion especially uses it to decompose a sparse matrix into two non sparse, low rank matrices which can then be used to predict…

We start with elementary algebraic theory of factorization of linear ordinary differential equations developed in the period 1880-1930. After exposing these classical results we sketch more sophisticated algorithmic approaches developed in…

We present an approach to decomposition and factor analysis of matrices with ordinal data. The matrix entries are grades to which objects represented by rows satisfy attributes represented by columns, e.g. grades to which an image is red, a…

Matrix factorization is a simple and effective solution to the recommendation problem. It has been extensively employed in the industry and has attracted much attention from the academia. However, it is unclear what the low-dimensional…

A vector space is commonly defined as a set that satisfies several conditions related to addition and scalar multiplication. However, for beginners, it may be hard to immediately grasp the essence of these conditions. There are probably a…

Given a fixed matrix, the problem of column subset selection requests a column submatrix that has favorable spectral properties. Most research from the algorithms and numerical linear algebra communities focuses on a variant called…

Let P be a commutative Noetherian ring, K be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and Pbar be the hypersurface ring P/(f). Assume that K:f is a grade four Gorenstein ideal of P.…

The increasing size of transformer-based models in NLP makes the question of compressing them important. In this work, we present a comprehensive analysis of factorization based model compression techniques. Specifically, we focus on…

The alternate row and column scaling algorithm applied to a positive $n\times n$ matrix $A$ converges to a doubly stochastic matrix $S(A)$, sometimes called the \emph{Sinkhorn limit} of $A$. For every positive integer $n$, a two parameter…

When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete…

We present an algorithm to reduce the computational effort for the multiplication of a given matrix with an unknown column vector. The algorithm decomposes the given matrix into a product of matrices whose entries are either zero or integer…

We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a…

When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being…

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 a data matrix $M$ and a…

An $r$-matrix is a matrix with symbols in $\{0,1,\dots,r-1\}$. A matrix is simple if it has no repeated columns. Let the support of a matrix $F$, $\text{supp}(F)$ be the largest simple matrix such that every column in $\text{supp}(F)$ is in…

Despite the popularity of factor models with sparse loading matrices, little attention has been given to formally address identifiability of these models beyond standard rotation-based identification such as the positive lower triangular…

This article studies how to form CUR decompositions of low-rank matrices via primarily random sampling, though deterministic methods due to previous works are illustrated as well. The primary problem is to determine when a column submatrix…