Computing all border bases for ideals of points

Hashemi, Amir; Kreuzer, Martin; Pourkhajouei, Samira

doi:10.1142/s0219498819501020

Cited by 5 publications

(4 citation statements)

References 17 publications

(51 reference statements)

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“…For example, Sauer [16] proposed an algorithm to compute approximate H bases, which consist of homogeneous polynomials. Hashemi et al [21] proposed a method to compute border bases for all possible monomial orders.…”

Section: Related Work a Monomial-based Algorithmsmentioning

confidence: 99%

Spurious Vanishing Problem in Approximate Vanishing Ideal

Kera

Hasegawa

2019

IEEE Access

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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 ideals plays a critical role. However, such an approximation also gives rise to a theoretical problem-the spurious vanishing problem-in the basis construction of approximate vanishing ideals; namely, obtained basis polynomials can be approximately vanishing simply because of the small coefficients. In this paper, we propose a first general method that enables various basis construction algorithms to overcome the spurious vanishing problem. In particular, we integrate coefficient normalization with polynomial-based basis constructions, which do not need the proper ordering of monomials to process for basis constructions. We further propose a method that takes advantage of the iterative nature of basis construction so that computationally costly operations for coefficient normalization can be circumvented. Moreover, a coefficient truncation method is proposed for further accelerations. From the experiments, it can be shown that the proposed method overcomes the spurious vanishing problem, resulting in shorter feature vectors while sustaining comparable or even lower classification error.INDEX TERMS Approximate commutative algebra, approximate vanishing ideal, machine learning.

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Section: Related Work a Monomial-based Algorithmsmentioning

confidence: 99%

Spurious Vanishing Problem in Approximate Vanishing Ideal

Kera

Hasegawa

2019

IEEE Access

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“…En general, cuando se trabaja con más variables y sin órdenes monomiales, los ideales monomiales asociados a un ideal fijo son llamados "border ideals" los cuales son un problema interesante por sí mismos. Sus aplicaciones pueden ser usadas en problemas de programación lineal, teoría de códigos, etc [11].…”

Section: Pantaleón-mondragón Petra Rubíunclassified

Border ideals: un algoritmo de pertenencia

Mondragon

2022

sahuarus

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La teoría de bases de Gröbner nos permite asociar a cualquier ideal en un anillo de polinomios con coeficientes en un campo un ideal monomial a través de un orden monomial, esto con el objetivo de obtener propiedades del ideal original a través del ideal monomial. Sin embargo, existen ideales monomiales asociados a un ideal dado que no provienen de un orden monomial. En este artículo, nos centraremos a estudiar y dar un algoritmo que nos permite determinar todos los ideales monomiales asociados a un ideal 0-dimensional dado en el anillo de polinomios en dos variables.

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“…Kreuzer 2006), most algorithms of the approximate vanishing ideal in computer algebra efficiently sidestep the issues with the spurious vanishing problem and basis set redundancy using the monomial order and symbolic computation. To our knowledge, there are two algorithms that work without the monomial order in computer algebra (Sauer 2007;Hashemi, Kreuzer, and Pourkhajouei 2019), but both require exponential-time procedures. Although the gradient has been rarely considered in the basis construction of the (approximate) vanishing ideal, Fassino (2010) used the gradient during basis construction to check whether a given polynomial exactly vanishes after slightly perturbing given points.…”

Section: Related Workmentioning

confidence: 99%

Gradient Boosts the Approximate Vanishing Ideal

Kera

Hasegawa

2020

AAAI

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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.

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Computing all border bases for ideals of points

Cited by 5 publications

References 17 publications

Spurious Vanishing Problem in Approximate Vanishing Ideal

Spurious Vanishing Problem in Approximate Vanishing Ideal

Border ideals: un algoritmo de pertenencia

Gradient Boosts the Approximate Vanishing Ideal

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