Approximate computation of zero-dimensional polynomial ideals

Heldt, Daniel; Kreuzer, Martin; Pokutta, Sebastian; Poulisse, H.N.J.

doi:10.1016/j.jsc.2008.11.010

Cited by 53 publications

(91 citation statements)

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“…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]. Corless et al raised a singular value decomposition (SVD) [22] approach for polynomial system and used it to solve the maximum common divisor problem [23], while SVD is the main step for solving the approximate vanishing ideal.…”

Section: Vca Methodsmentioning

confidence: 99%

“…Stetter developed the theory proposed by Corless et al and presented a more general numerical method [24]. Heldt et al utilized SVD and stable numerical method [25] to solve the approximate vanishing ideal, and these vanishing component polynomials almost composed a border basis [21]. Heldt et al also worked out the Cohen-Macaulay basis of vanishing ideal [26].…”

Section: Vca Methodsmentioning

confidence: 99%

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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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Section: Vca Methodsmentioning

confidence: 99%

Section: Vca Methodsmentioning

confidence: 99%

Improvement on the vanishing component analysis by grouping strategy

Zhang

2018

J Wireless Com Network

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“…e.g., [47]) and they have a variety of applications, e.g., in open-pit mining where any feasible production plan is indeed an order ideal; we refer the interested reader to [30] for an overview. Another example is the approximate vanishing ideal algorithm in [29], which computes a polynomial description of an approximate vanishing ideal of a given set of (noisy) points. Effectively, a total-least-squares optimization problem is solved here and explanatory variables can come from an order ideal.…”

Section: Arbitrary Order Idealsmentioning

confidence: 99%

“…An example illustrating these two cases is presented in [35,Example 6]. Finally, border bases deform more smoothly in the input [40] (see also [49]), which is particularly helpful when the coefficients arise from measurement data [1,29], e.g., that is why border bases are used in the context of total-least-squares polynomial regression (see [29] for details).…”

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confidence: 99%

A Polyhedral Characterization of Border Bases

Braun¹,

Pokutta²

2016

SIAM J. Discrete Math.

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A. Border bases arise as a canonical generalization of Gröbner bases, using order ideals instead of term orderings. We provide a polyhedral characterization of all order ideals (and hence all border bases) that are supported by a zero-dimensional ideal: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. In particular, we establish a crucial connection between the ideal and its combinatorial structure. Based on this characterization we also provide an adaptation of the border basis algorithm of Kehrein and Kreuzer [35] to allow for computing border bases for arbitrary order ideals, given implicitly via maximizing a preference on monomials (variable selection problem), independent of term orderings. The algorithm requires the same size of resources as the border basis algorithm except for some minor overhead. We also show that the underlying variable selection problem of finding an order ideal that supports a border basis is NP-hard and that any linear description of the associated convex hull of all order ideals requires a superpolynomial number of inequalities. IIn many different disciplines and real-world applications one is faced with solving systems of polynomial equations. Often this is simply due to a physical or dynamical system having a natural representation as a system of polynomial equations, but equally often it is due to the sheer expressive power of polynomial systems of equations that allow for easy reformulation. To give an example of the latter, an inequality ax ≤ b with a ∈ R n and b ∈ R can be expressed via a single polynomial equation: ax + u 2 = b. A slightly more involved example is that of the feasible region of a binary program {x | Ax ≤ b, x ∈ {0, 1} n }, which can be captured via rewriting each individual inequality as before, and adding quadratic polynomials x 2 i − x i = 0 for each coordinate i = 1, . . . , n of x. As a consequence, there is a huge need to computationally model, understand, manipulate, and extract the solution set of systems of polynomial equations.A key insight in (computational) commutative algebra is that one can choose a smart ordering on the monomials and compute a special set of generators of the ideal generated by the system of equations that makes many operations easy, and provides a structural insight into the system. One such special set of generators is a Gröbner basis. By now, Gröbner bases are fundamental and standard tools in commutative algebra to actually perform important operations on ideals such as intersection, membership test, elimination, projection, and many more. Border bases arise as a natural generalization of Gröbner bases that can be computed for zero-dimensional ideals, i.e., the associated factor space is a finite-dimensional vector space (see Section 2 for details). While this might seem to be a severe restriction, for many applications it is sufficient. Roughly speaking, whenever the solution set is finite, we are dealing with a zero-dimensional ideal. For example, systems of p...

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“…An interesting class of recently developed algorithms relies on tools from Numerical Commutative Algebra [17,2,10,6,7]. For all these algorithms the input is a set of points possibly in n-dimensions and the output is a polynomial f in n-variables whose zero locus (which is a curve, or a surface, or more generally an algebraic variety) gives an approximation for the input points and can be interpreted as an implicit polynomial regression model [12,Ch 2].…”

Section: Step I: Approximation Of a Path By A Polynomial Curvementioning

confidence: 99%

An Euclidean norm based criterion to assess robots’ 2D path-following performance

Saggini

Torrente

2016

J Alg Stat

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Abstract. A current need in the robotics field is the definition of methodologies for quantitatively evaluating the results of experiments. This paper contributes to this by defining a new criterion for assessing path-following tasks in the planar case, that is, evaluating the performance of robots that are required to follow a desired reference path. Such criterion comes from the study of the local differential geometry of the problem. New conditions for deciding whether or not the zero locus of a given polynomial intersects the neighbourhood of a point are defined. Based on this, new algorithms are presented and tested on both simulated data and experiments conducted at sea employing an Unmanned Surface Vehicle.2000 Mathematics Subject Classifications: 26C10, 15A60, 93C85, 62P99.

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Approximate computation of zero-dimensional polynomial ideals

Cited by 53 publications

References 10 publications

Improvement on the vanishing component analysis by grouping strategy

Improvement on the vanishing component analysis by grouping strategy

A Polyhedral Characterization of Border Bases

An Euclidean norm based criterion to assess robots’ 2D path-following performance

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