The construction of multivariate polynomials with preassigned zeros

Möller, H. Michael; Buchberger, Bruno

doi:10.1007/3-540-11607-9_3

Cited by 101 publications

(83 citation statements)

References 3 publications

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“…As mentioned before, this problem appears not to be very exciting at first view, as for any finite set of points there exists a set F of polynomials with the property that f ( ) = 0 if and only if f ∈ F . How to find a basis of the ideal I ( ), more precisely, even a Gröbner basis of the ideal, has already been described in [10]. Nevertheless, intuitively the problem of detecting whether is contained in some algebraic variety, has a slightly different flavor as the following simple example indicates.…”

Section: Notation and Preliminariesmentioning

confidence: 94%

“…In other words, the problem could be rephrased as finding a set F such that ⊆ V(F). However, put this way, the answer to the question is well-known: whenever is a finite set, it defines a zero dimensional ideal I ( ) in the ring of polynomials and thus there always exists a set F of polynomials such that F( ) = 0, for example a Gröbner basis for I ( ), which can be constructed from by means of the Buchberger-Möller algorithm, see [10]. The construction is such that I ( ) is generated by F and therefore = V (F), the ideal is the smallest one whose variety contains .…”

Section: Introductionmentioning

confidence: 99%

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Approximate varieties, approximate ideals and dimension reduction

Sauer

2007

Numer Algor

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Section: Notation and Preliminariesmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Approximate varieties, approximate ideals and dimension reduction

Sauer

2007

Numer Algor

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“…As data often contains noises, it is difficult to acquire an analytical solution, so numerical methods are needed to solve an approximate vanishing ideal. The VCA method-related research involves as follows: Buchberger and Möller et al firstly proposed an algorithm to figure out the vanishing ideal of finite point set, called BuchbergerMöller algorithm [20], which can be regarded as the Euclidean algorithm that solves the maximum common divisor of single variable and the generalization of Gaussian elimination method in a linear system. The obtained Grobner basis has stable values when the coordinate system is measurable [21].…”

Section: Vca Methodsmentioning

confidence: 99%

Improvement on the vanishing component analysis by grouping strategy

Zhang

2018

J Wireless Com Network

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Vanishing component analysis (VCA) method, as an important method integrating commutative algebra with machine learning, utilizes the polynomial of vanishing component to extract the features of manifold, and solves the classification problem in ideal space dual to kernel space. But there are two problems existing in the VCA method: first, it is difficult to set a threshold of its classification decision function. Second, it is hard to handle with the over-scaled training set and oversized dimension of eigenvector. To address these two problems, this paper improved the VCA method and presented a grouped VCA (GVCA) method by grouping strategy. The classification decision function did not use a predetermined threshold; instead, it solved the values of all polynomials of vanishing component and sorted them, and then used majority voting approach to determine their classes. After that, a strategy of grouping training set was proposed to segment training sets into multiple non-intersecting subsets, which polynomials of vanishing component were later acquired through a VCA method, respectively, and finally combined into an integral set of vanishing component polynomial. What is more important is that it uses the bagging theory in ensemble learning to successfully expound and prove the correctness of the strategy of grouping training sets. It also compares the time complexity for training algorithm with and without grouping training sets, thus demonstrating the effectiveness of the grouping strategy. A series of experiments showed that the GVCA method proposed in the paper has a perfect classification performance with a rapid rate of convergence compared to other statistical learning methods.

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“…The NBM algorithm is from the field of approximate computational algebraic geometry and is based on a least square approximation. It is a variation of a purely symbolic algorithm: the Buchberger-Möller algorithm (Möller and Buchberger [1982]) and its spirit is numerical.…”

Section: Motivations For a Numerical Fan Of A Designmentioning

confidence: 99%