相关论文: On tropical and nonnegative factorization ranks of…
In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call `tropical polynomials' and `sign polynomials', respectively. We classify all irreducible polynomials in either case. We…
This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain…
A novel approach to Boolean matrix factorization (BMF) is presented. Instead of solving the BMF problem directly, this approach solves a nonnegative optimization problem with the constraint over an auxiliary matrix whose Boolean structure…
Matrix representations are a powerful tool for designing efficient algorithms for combinatorial optimization problems such as matching, and linear matroid intersection and parity. In this paper, we initiate the study of matrix…
In this paper, we investigate the computational complexity of the knapsack problem and subset sum problem for the following tropical algebraic structures. We consider the semigroup of square matrices of size $k \times k$ with non-negative…
We prove the conjecture that, for any $n$, the monoid of all $n \times n$ tropical matrices satisfies nontrivial semigroup identities. To this end, we prove that the factor rank of a large enough power of a tropical matrix does not exceed…
We investigate compressibility of the dimension of positive semidefinite matrices while approximately preserving their pairwise inner products. This can either be regarded as compression of positive semidefinite factorizations of…
We derive approximation algorithms for the nonnegative matrix factorization problem, i.e. the problem of factorizing a matrix as the product of two matrices with nonnegative coefficients. We form convex approximations of this problem which…
The nonnegative integer rank of a matrix is a variant of the classical nonnegative rank, introduced in the 1980s, where factorizations are required to have integer entries. While computing nonnegative integer rank is generally very hard, we…
We consider constrained optimization problems defined in the tropical algebra setting on a linearly ordered, algebraically complete (radicable) idempotent semifield (a semiring with idempotent addition and invertible multiplication). The…
Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known,…
Satisfiability of Boolean circuits is among the most known and important problems in theoretical computer science. This problem is NP-complete in general but becomes polynomial time when restricted either to monotone gates or linear gates.…
Matrix factorization is a popular approach for large-scale matrix completion. The optimization formulation based on matrix factorization can be solved very efficiently by standard algorithms in practice. However, due to the non-convexity…
This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the…
Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of…
It is well-known that the Burer-Monteiro (B-M) factorization approach can efficiently solve low-rank matrix optimization problems under the RIP condition. It is natural to ask whether B-M factorization-based methods can succeed on any…
In this paper we study the complexity of factorization of polynomials in the free noncommutative ring $\mathbb{F}\langle x_1,x_2,\dots,x_n\rangle$ of polynomials over the field $\mathbb{F}$ and noncommuting variables $x_1,x_2,\ldots,x_n$.…
We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in…
Triangular factorizations are an important tool for solving integral equations and partial differential equations with hierarchical matrices ($\mathcal{H}$-matrices). Experiments show that using an $\mathcal{H}$-matrix LR factorization to…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…