Related papers: Border basis computation with gradient-weighted no…

This paper studies the concept and the computation of approximately vanishing ideals of a finite set of data points. By data points, we mean that the points contain some uncertainty, which is a key motivation for the approximate treatment.…

In the last decade, the approximate basis computation of vanishing ideals has been studied extensively in computational algebra and data-driven applications such as machine learning. However, symbolic computation and the dependency on term…

In the last decade, the approximate vanishing ideal and its basis construction algorithms have been extensively studied in computer algebra and machine learning as a general model to reconstruct the algebraic variety on which noisy data…

Approximate vanishing ideal is a concept from computer algebra that studies the algebraic varieties behind perturbed data points. To capture the nonlinear structure of perturbed points, the introduction of approximation to exact vanishing…

Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable,…

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced…

This paper describes and analyzes a method for computing border bases of a zero-dimensional ideal $I$. The criterion used in the computation involves specific commutation polynomials and leads to an algorithm and an implementation extending…

In this paper, we generalize the notion of border bases of zero-dimensional polynomial ideals to the module setting. To this end, we introduce order modules as a generalization of order ideals and module border bases of submodules with…

Border bases arise as a canonical generalization of Gr\"obner bases. We provide a polyhedral characterization of all order ideals (and hence border bases) that are supported by a zero-dimensional ideal: order ideals that support a border…

Solving systems of polynomial equations, particularly those with finitely many solutions, is a crucial challenge across many scientific fields. Traditional methods like Gr\"obner and Border bases are fundamental but suffer from high…

Deep neural networks have reshaped modern machine learning by learning powerful latent representations that often align with the manifold hypothesis: high-dimensional data lie on lower-dimensional manifolds. In this paper, we establish a…

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…

In this paper, we describe a new method to compute the minimum of a real polynomial function and the ideal defining the points which minimize this polynomial function, assuming that the minimizer ideal is zero-dimensional. Our method is a…

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finite. The aim is to combine approaches for solving a system of polynomial equations with dual…

Although it is relatively easy to apply, the gradient method often displays a disappointingly slow rate of convergence. Its convergence is specially based on the structure of the matrix of the algebraic linear system, and on the choice of…

In this paper we introduce a working generalization of the theory of Gr\"obner bases for algebras of partial difference polynomials with constant coefficients. One obtains symbolic (formal) computation for systems of linear or non-linear…

We consider the computation of averaged coefficients for the homogenization of elliptic partial differential equations. In this problem, like in many multiscale problems, a large number of similar computations parametrized by the…

A computation method of algebraic local cohomology with parameters, associated with zero-dimensional ideal with parameter, is introduced. This computation method gives us in particular a decomposition of the parameter space depending on the…

In geostatistics, traditional spatial models often rely on the Gaussian Process (GP) to fit stationary covariances to data. It is well known that this approach becomes computationally infeasible when dealing with large data volumes,…

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…