Related papers: Conditional Gradients for the Approximate Vanishin…

Let $\mathbb{F}[X]$ be the polynomial ring over the variables $X=\{x_1,x_2, \ldots, x_n\}$. An ideal $I=\langle p_1(x_1), \ldots, p_n(x_n)\rangle$ generated by univariate polynomials $\{p_i(x_i)\}_{i=1}^n$ is a \emph{univariate ideal}. We…

For an ideal $I\subseteq\mathbb{R}[x]$ given by a set of generators, a new semidefinite characterization of its real radical $I(V_\mathbb{R}(I))$ is presented, provided it is zero-dimensional (even if $I$ is not). Moreover we propose an…

Let $\K$ be a finite field and $X$ be a complete simplicial toric variety over $\K$. We give an algorithm relying on elimination theory for finding generators of the vanishing ideal of a subgroup $Y_Q$ parameterized by a matrix $Q$ which…

We study the arithmetic complexity of hitting set generators, which are pseudorandom objects used for derandomization of the polynomial identity testing problem. We give new explicit constructions of hitting set generators whose outputs are…

Let C be an irreducible projective curve of degree d in Pn(K), where K is an algebraically closed field, and let I be the associated homogeneous prime ideal. We wish to compute generators for I, assuming we are given sufficiently many…

Let R be a standard graded polynomial ring in f variables over a field and Psi be an f by g matrix of linear forms from R, where g is positive and less than f. Assume that the row vector of variables annihilates Psi and that the ideal I…

$ \newcommand{\ie}{i.\,e.} $We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to…

This paper introduces the Nystr\"om PCG algorithm for solving a symmetric positive-definite linear system. The algorithm applies the randomized Nystr\"om method to form a low-rank approximation of the matrix, which leads to an efficient…

Given a finite set of closed rational points of affine space over a field, we give a Gr\"obner basis for the lexicographic ordering of the ideal of polynomials which vanish at all given points. Our method is an alternative to the…

Policy gradients methods apply to complex, poorly understood, control problems by performing stochastic gradient descent over a parameterized class of polices. Unfortunately, even for simple control problems solvable by standard dynamic…

We present an encoding of a polynomial system into vanishing and non-vanishing constraints on almost-principal minors of a symmetric, principally regular matrix, such that the solvability of the system over some field is equivalent to the…

We provide an algorithm that computes a set of generators for any complete ideal in a smooth complex surface. More interestingly, these generators admit a presentation as monomials in a set of maximal contact elements associated to the…

In modeling physical systems it is sometimes useful to construct border bases of 0-dimensional polynomial ideals which are contained in the ideal generated by a given set of polynomials. We define and construct such subideal border bases,…

In this paper we characterize, in algebraic and geometric terms, when a graded vanishing ideal is generated by binomials over any field K.

A subgradient method is presented for solving general convex optimization problems, the main requirement being that a strictly-feasible point is known. A feasible sequence of iterates is generated, which converges to within user-specified…

We consider an homogeneous ideal $I$ in the polynomial ring $S=K[x_1,\dots,$ $x_m]$ over a finite field $K=\mathbb{F}_q$ and the finite set of projective rational points $\mathbb{X}$ that it defines in the projective space…

We construct an explicit minimal strong Groebner basis of the ideal of vanishing polynomials in the polynomial ring over Z/m for m>=2. The proof is done in a purely combinatorial way. It is a remarkable fact that the constructed Groebner…

Recently, a series of Image-Text Matching (ITM) methods achieve impressive performance. However, we observe that most existing ITM models suffer from gradients vanishing at the beginning of training, which makes these models prone to…

The study of first-order optimization is sensitive to the assumptions made on the objective functions. These assumptions induce complexity classes which play a key role in worst-case analysis, including the fundamental concept of algorithm…

Graph data structures are fundamental for studying connected entities. With an increase in the number of applications where data is represented as graphs, the problem of graph generation has recently become a hot topic. However, despite its…