7 research outputs found
Corrigendum for "Almost vanishing polynomials and an application to the Hough transform"
In this note we correct a technical error occurred in [M. Torrente and M.C.
Beltrametti, "Almost vanishing polynomials and an application to the Hough
transform", J. Algebra Appl. 13(8), (2014)]. This affects the bounds given in
that paper, even though the structure and the logic of all proofs remain fully
unchanged.Comment: 30 page
Gradient Boosts the Approximate Vanishing Ideal
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 approximately lie. In particular, the basis construction
algorithms developed in machine learning are widely used in applications across
many fields because of their monomial-order-free property; however, they lose
many of the theoretical properties of computer-algebraic algorithms. In this
paper, we propose general methods that equip monomial-order-free algorithms
with several advantageous theoretical properties. Specifically, we exploit the
gradient to (i) sidestep the spurious vanishing problem in polynomial time to
remove symbolically trivial redundant bases, (ii) achieve consistent output
with respect to the translation and scaling of input, and (iii) remove
nontrivially redundant bases. The proposed methods work in a fully numerical
manner, whereas existing algorithms require the awkward monomial order or
exponentially costly (and mostly symbolic) computation to realize properties
(i) and (iii). To our knowledge, property (ii) has not been achieved by any
existing basis construction algorithm of the approximate vanishing ideal.Comment: 9+10 pages, 1+4 figures, AAAI'2
Zero-dimensional families of polynomial systems
If a real world problem is modelled with a system of polynomial equations, the coefficients are in general not exact. The consequence is that small perturbations of the coefficients may lead to big changes of the solutions. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which the family of all ideals shares the number of isolated real zeros. In the second part we show how to modify the equations and get new ones which generate the same ideal, but whose real zeros are more stablewith respect to perturbations of the coefficients
Simple Varieties for Limited Precision Points
Given a finite set X of points and a tolerance epsilon representing the maximum error on the coordinates of each point, we address the problem of computing a simple polynomial f whose zero-locus Z(f) ``almost'' contains the points of X.
We propose a symbolic-numerical method that, starting from the knowledge of X and epsilon, determines a polynomial f whose degree is strictly bounded by the minimal degree of the lements of the vanishing ideal of X. Then we state the sufficient conditions for proving that Z(f) lies close to each point of X by less than epsilon. The validity of the proposed method relies on a combination of classical results of Computer Algebra and Numerical Analysis; its effectiveness is
illustrated with a number of examples