相关论文: An algorithm to describe the solution set of any t…

In this paper, we consider the solution of ill-conditioned systems of linear algebraic equations that can be determined imprecisely. To improve the stability of the solution process, we "immerse" the original imprecise linear system in an…

We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and…

In the last few years there has been a growing interest towards methods for statistical inference and learning based on computational geometry and, notably, tropical geometry, that is, the study of algebraic varieties over the min-plus…

In this paper, we investigate tropical secant varieties of ordinary linear spaces. These correspond to the log-limit sets of ordinary toric varieties; we show that their interesting parts are combinatorially isomorphic to a certain natural…

This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…

Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for…

We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…

In this paper, a multiplicity preserving triangular set decomposition algorithm is proposed for a system of two polynomials. The algorithm decomposes the variety defined by the polynomial system into unmixed components represented by…

Bilevel optimization has been widely used in decision-making process. However, there still lacks an efficient algorithm to determine an optimal solution of a bilevel optimization problem, especially for a large-size problem. To bridge the…

Tropical algebraic geometry offers new tools for elimination theory and implicitization. We determine the tropicalization of the image of a subvariety of an algebraic torus under any homomorphism from that torus to another torus.

An arrangement of finitely many tropical hyperplanes in the tropical torus leads to a notion of `type' data for points, with the underlying unlabeled arrangement giving rise to `coarse type'. It is shown that the decomposition of the…

The linear system $|D|$ of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all…

We consider the problem of solving a linear system of equations which involves complex variables and their conjugates. We characterize when it reduces to a complex linear system, that is, a system involving only complex variables (and not…

We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major…

This article presents a methodology that automatically derives a combinatorial specification for a permutation class C, given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite. This is…

The construction of the general solution sequence of row-finite linear systems is accomplished by implementing -ad infinitum- the Gauss-Jordan algorithm under a rightmost pivot elimination strategy. The algorithm generates a basis (finite…

We consider a class of linear programs on graphs with total variation regularization and a budgetary constraint. For these programs, we give a characterization of basic solutions in terms of rooted spanning forests with orientation on the…

This paper presents iterative methods for solving tensor equations involving the T-product. The proposed approaches apply tensor computations without matrix construction. For each initial tensor, these algorithms solve related problems in a…

We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…