Related papers: A universality theorem for nonnegative matrix fact…
A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set of generators and relations for the semigroup of $k$-nonnegative $n\times n$ invertible matrices in two special cases: when $k =…
Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…
We introduce categories of weak factorization algebras and factorization spaces, and prove that they are equivalent to the categories of ordinary factorization algebras and spaces, respectively. This allows us to define the pullback of a…
A problem by Feichtinger, Heil, and Larson asks whether every infinite matrix $A$ with $\sum_{k,l}|A_{kl}| < \infty$ (an equivalent substitute for the Feichtinger algebra) that is positive-semidefinite admits a symmetric rank-one…
Theory of matrix factorizations is useful to study hypersurfaces in commutative algebra. To study noncommutative hypersurfaces, which are important objects of study in noncommutative algebraic geometry, we introduce a notion of…
In 2005, Boman et al introduced the concept of factor width for a real symmetric positive semidefinite matrix. This is the smallest positive integer $k$ for which the matrix $A$ can be written as $A=VV^T$ with each column of $V$ containing…
We present a non-standard proof of the fact that the existence of a local (i.e. restricted to a point) characteristic-zero, semi-parametric lifting for a variety defined by the zero locus of polynomial equations over the integers is…
A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A=BB^T. If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A. In…
Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of…
It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive…
In a nutshell, we show that polynomials and nested polytopes are topological, algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a polytope…
The rank of an n x n matrix A is equal to the size of its largest square submatrix with a nonzero determinant, and it can be computed in O(n^2.37) time. Analogously, the size of the largest square submatrix with nonzero permanent is defined…
A subspace of the space, L(n), of traceless complex $n\times n$ matrices can be specified by requiring that the entries at some positions $(i,j)$ be zero. The set, $I$, of these positions is a (zero) pattern and the corresponding subspace…
We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic…
This paper considers the problem of positive semidefinite factorization (PSD factorization), a generalization of exact nonnegative matrix factorization. Given an $m$-by-$n$ nonnegative matrix $X$ and an integer $k$, the PSD factorization…
In this work, we investigate the existence of a factorization for a unital completely positive map, between non-commutative probability space which do not change the expectation values of the events. These maps are called in literature…
Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative $n \times m$ matrix $M$ into a product of a nonnegative $n \times d$ matrix $W$ and a nonnegative $d \times m$ matrix $H$. A longstanding open…
For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix…