Related papers: Minimal rank factorizations of polynomial matrices
Low rank matrix factorization is a fundamental building block in machine learning, used for instance to summarize gene expression profile data or word-document counts. To be robust to outliers and differences in scale across features, a…
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $\mathbb{R}^{n \times…
Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…
Recommendation systems often rely on point-wise loss metrics such as the mean squared error. However, in real recommendation settings only few items are presented to a user. This observation has recently encouraged the use of rank-based…
The classic rank-revealing QR factorization factorizes a matrix $A$ as $AP=QR$ where $P$ permutes the columns of $A$, $Q$ is an orthogonal matrix, and $R$ is upper triangular with non-increasing diagonal entries. This is called…
Differentiable systems in this paper means systems of equations that are described by differentiable real functions in real matrix variables. This paper proposes algorithms for finding minimal rank solutions to such systems over (arbitrary…
Block full rank pencils introduced in [Dopico et al., Local linearizations of rational matrices with application to rational approximations of nonlinear eigenvalue problems, Linear Algebra Appl., 2020] allow us to obtain local information…
It is known that the common factors in a large panel of data can be consistently estimated by the method of principal components, and principal components can be constructed by iterative least squares regressions. Replacing least squares…
This paper investigates atomic factorizations in the monoid $\mathcal I(R)$ of nonzero ideals of a multivariate polynomial ring $R$, under ideal multiplication. Building on recent advances in factorization theory for unit-cancellative…
Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and…
We study the problem of constructing explicit families of matrices which cannot be expressed as a product of a few sparse matrices. In addition to being a natural mathematical question on its own, this problem appears in various…
We consider the problem of computing matrix polynomials $p(X)$, where $X$ is a large dense matrix, with as few matrix-matrix multiplications as possible. More precisely, let $\Pi_{2^{m}}^*$ represent the set of polynomials computable with…
In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity…
Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…
This article characterizes the rank-one factorization of auto-correlation matrix polynomials. We establish a sufficient and necessary uniqueness condition for uniqueness of the factorization based on the greatest common divisor (GCD) of…
It is well known that any nonsingular M-matrix admits an LU factorization into M-matrices (with L and U lower and upper triangular respectively) and any singular M-matrix is permutation similar to an M-matrix which admits an LU…
We study the complexity of solving the \emph{generalized MinRank problem}, i.e. computing the set of points where the evaluation of a polynomial matrix has rank at most $r$. A natural algebraic representation of this problem gives rise to a…
Low rank approximation is a commonly occurring problem in many computer vision and machine learning applications. There are two common ways of optimizing the resulting models. Either the set of matrices with a given rank can be explicitly…
We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…
Nowadays, the availability of large-scale data in disparate application domains urges the deployment of sophisticated tools for extracting valuable knowledge out of this huge bulk of information. In that vein, low-rank representations…