Related papers: Newton polytopes and numerical algebraic geometry
We show how to construct highly symmetric algorithms for matrix multiplication. In particular, we consider algorithms which decompose the matrix multiplication tensor into a sum of rank-1 tensors, where the decomposition itself consists of…
We introduce an algorithm of joint approximation of a function and its first derivative by alternative orthogonal polynomials on the interval [0,1].The algorithm exhibits properties of shape preserving approximation for the function. A weak…
In numerical linear algebra, a well-established practice is to choose a norm that exploits the structure of the problem at hand in order to optimize accuracy or computational complexity. In numerical polynomial algebra, a single norm…
Numerical Algebraic Geometry uses numerical data to describe algebraic varieties. It is based on the methods of numerical polynomial homotopy continuation, an alternative to the classical symbolic approaches of computational algebraic…
When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer…
This paper investigates the cost of solving systems of sparse polynomial equations by homotopy continuation. First, a space of systems of $n$-variate polynomial equations is specified through $n$ monomial bases. The natural locus for the…
Motivated by a connection with the factorization of multivariate polynomials, we study integral convex polytopes and their integral decompositions in the sense of the Minkowski sum. We first show that deciding decomposability of integral…
In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we…
This paper proposes and develops a new Newton-type algorithm to solve subdifferential inclusions defined by subgradients of extended-real-valued prox-regular functions. The proposed algorithm is formulated in terms of the second-order…
This article is an introduction to a new approach to first principles electronic structure calculation. The starting point is the Hartree-Fock-Roothaan equation, in which molecular integrals are approximated by polynomials by way of Taylor…
Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral…
Finding roots of equations is at the heart of most computational science. A well-known and widely used iterative algorithm is the Newton's method. However, its convergence depends heavily on the initial guess, with poor choices often…
We propose a gradient descent method for solving optimization problems arising in settings of tropical geometry - a variant of algebraic geometry that has attracted growing interest in applications such as computational biology, economics,…
Polytope numbers for a polytope are a sequence of nonnegative integers that are defined by the facial information of a polytope. Every polygon is triangulable and a higher dimensional analogue of this fact states that every polytope is…
We illustrate an efficient new method for handling polynomial systems with degenerate solution sets. In particular, a corollary of our techniques is a new algorithm to find an isolated point in every excess component of the zero set (over…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
We present numerical homotopy continuation algorithms for solving systems of equations on a variety in the presence of a finite Khovanskii basis. These take advantage of Anderson's flat degeneration to a toric variety. When Anderson's…
An efficient algorithm to enumerate the vertices of a two-dimensional (2D) projection of a polytope, is presented in this paper. The proposed algorithm uses the support function of the polytope to be projected and enumerated for vertices.…
We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in…
Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been…