Approximate varieties, approximate ideals and dimension reduction

Sauer, Tomas

doi:10.1007/s11075-007-9112-4

Cited by 22 publications

(25 citation statements)

References 9 publications

(7 reference statements)

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“…Heldt et al also worked out the Cohen-Macaulay basis of vanishing ideal [26]. Sauer et al made use of a strategy of independent coordinate and increased degree of polynomial to calculate the approximate vanishing ideal, thereby acquiring an approximate solution of Buchberger-Möller algorithm [27]. Kiral et al raised two dualities, namely, the duality between kernel and ideal and the duality between ideal and manifold pattern.…”

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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Section: Vca Methodsmentioning

confidence: 99%

Improvement on the vanishing component analysis by grouping strategy

Zhang

2018

J Wireless Com Network

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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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“…Consider the case s = q = 2, and the moment array (39). We show only the elements in (2,4) = (2,2) + (2,2) . Then we have that [1,1] .…”

Section: Construction Of Shifted Moment Arraysmentioning

confidence: 99%

“…Fitting two-dimensional data by conic sections (q = 2) is the most common case of algebraic hypersurface fitting, with numerous applications in robotics, medical imaging, archaeology, etc., see [1] for an overview. Fitting algebraic hypersurfaces of higher degrees and dimensions is needed in computer graphics [2], computer vision [3], and symbolic-numeric computations [4], [5, §5]. The problem also appears in advanced methods of multivariate data analysis such as subspace clustering [6] and non-linear system identification [7], see [8] for an overview.…”

Section: Introductionmentioning

confidence: 99%

Adjusted least squares fitting of algebraic hypersurfaces

Usevich

Markovsky

2016

Linear Algebra and its Applications

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We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.

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Approximate varieties, approximate ideals and dimension reduction

Abstract: We consider a method to determine, for a given finite set of points, an algebraic variety of small degree which contains these points. In contrast to most other algorithms in Computer Algebra, this one is adapted to numerical, inexact computations.

Cited by 22 publications

References 9 publications

Improvement on the vanishing component analysis by grouping strategy

Improvement on the vanishing component analysis by grouping strategy

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

Adjusted least squares fitting of algebraic hypersurfaces

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