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SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 2008年第11期18卷 3409-3430页

作者： Ginoux, Jean-Marc Rossetto, Bruno Chua, Leon O. Univ Sud Lab PROTEE IUT Toulon F-83957 La Garde France Univ Calif Berkeley Dept EECS Berkeley CA 94720 USA

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chua's piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

关键词： Differential geometry curvature torsion gram-schmidt algorithm Darboux invariant

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How many entries of a typical orthogonal matrix can be approximated by independent normals?

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ANNALS OF PROBABILITY 2006年第4期34卷 1497-1529页

作者： Jiang, Tiefeng Univ Minnesota Minneapolis MN 55455 USA

We solve an open problem of Diaconis that asks what are the largest orders of p(n) and q(n) such that Z(n), the p(n) x q(n) upper left block of a random matrix Gamma(n) which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the variation distance between the joint distribution of entries of Z(n) and that of p(n)q(n) independent standard normals goes to zero provided p(n) = o(root n) and q(n) = o(root n). We also show that the above variation distance does not go to zero if p(n) = [x root n] and q(n) = [y root n] for any positive numbers x and y. This says that the largest orders of p(n) and q(n) are o(n(1/2)) in the sense of the above approximation. Second, suppose Gamma(n) = (y(ij))(nxn) is generated by performing the gram-schmidt algorithm on the columns of Y-n = (y(ij))(nxn), where {y(ij);1 <= i, j <= n} are i.i.d. standard normals. We show that epsilon(n)(m) : = max(1 <= i <= n,1 <= j <= m)vertical bar root n center dot y(ij) - y(ij)vertical bar goes to zero in probability as long as m = m(n) = o(n/log n). We also prove that epsilon(n) (m(n)) -> 2 root alpha in probability when m(n) = [n alpha/log n] for any alpha > 0. This says that m(n) = o(n/log n) is the largest order such that the entries of the first m(n) columns of Gamma(n) can be approximated simultaneously by independent standard normals.

关键词： Haar measure gram-schmidt algorithm large deviation maxima product distribution random matrix theory variation distance

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New Learning Approach for Unsupervised Neural Networks Model with Application to Agriculture Field

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INTERNATIONAL JOURNAL OF ADVANCED COMPUTER SCIENCE AND APPLICATIONS 2020年第5期11卷 360-369页

作者： Sara, Belattar Otman, Abdoun Haimoudi, El Khatir Univ Abdelmalek Essaadi UAE Comp Sci Dept Adv Sci & Technol Lab Polydisciplinary Fac Larache Morocco

An accurate and lower cost hybrid machine learning algorithm based on a combination of Kohonen-Self Organizing Map (SOM) and gram-schmidt (GSHM) algorithm was proposed, to enhance the crop yield prediction and to increase the agricultural production. The combination of GSHM and SOM allows to withdraw the most informative components about our data, by overcoming correlation issues between input data prior to the training process. The improved hybrid algorithm was trained firstly on data that have a correlation problem, and it was compared with another hybrid model based on SOM and Principal Component Analysis (PCA), secondly, it was trained using selected soil parameters related to the atmosphere (e.g. pH, nitrogen, phosphate, potassium, depth, temperature, and rainfall). A comparative study with the standard SOM was conducted. The improved Kohonen-Self Organizing Map when applied to correlated data, demonstrated better results in terms of classification accuracy (8/8), and rapidity = 0.015s compared to a classification accuracy (7/8) and a rapidity = 97,828 s using SOM combined with PCA. Moreover, the proposed algorithm resulted in better results for crop prediction in terms of maximum iteration number of 675, mean error <= 0.00022, and rapidity = 18.422s versus an iteration number of 729, mean error <= 0.000916 and rapidity= 23.707s with the standard SOM. The proposed algorithm allowed us to overcome correlation issues, and to improve the classification, learning process, and rapidity, with the potential to apply for predicting crop yield in the agricultural field.

关键词： Kohonen-self organizing map gram-schmidt algorithm principal component analysis agriculture field crop yield prediction

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BLOCK gram-schmidt ORTHOGONALIZATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2008年第1期31卷 761-775页

作者： Stewart, G. W. Univ Maryland Dept Comp Sci College Pk MD 20742 USA Univ Maryland Inst Adv Comp Studies College Pk MD 20742 USA

The classical gram-schmidt algorithm for computing the QR factorization of a matrix X requires at least one pass over the current orthogonalized matrix Q as each column of X is added to the factorization. When Q becomes so large that it must be maintained on a backing store, each pass involves the costly transfer of data from the backing store to main memory. However, if one orthogonalizes the columns of X in blocks of m columns, the number of passes is reduced by a factor of 1/m. Moreover, matrix-vector products are converted into matrix-matrix products, allowing level-3 BLAS cache performance. In this paper we derive such a block algorithm and give some experimental results that suggest it can be quite effective for large scale problems, even when the matrix X is rank degenerate.

关键词： orthogonalization gram-schmidt algorithm blocked algorithm QR factorization

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A Receding Horizon-type Solution of the Inverse Kinematic Task of Redundant Robots 15

A Receding Horizon-type Solution of the Inverse Kinematic Ta...

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IEEE 15th International Symposium on Applied Computational Intelligence and Informatics (SACI)

作者： Issa, Hazem Varga, Bence Tar, Jozsef K. Obuda Univ Doctoral Sch Appl Informat & Appl Math Budapest Hungary Obuda Univ Inst Mechatron & Vehicle Engn Donat Banki Fac Mech & Safety Engn Budapest Hungary Obuda Univ Antal Bejczy Ctr Intelligent Robot Univ Res & Innovat Ctr Budapest Hungary Obuda Univ John von Neumann Fac Informat Inst Appl Math Budapest Hungary

ISBN: (纸本)9781728195438

With the exception of specially designed kinematic structures, the inverse kinematic task of redundant robots cannot be solved in "closed form". Redundancy is often needed to make the arm "dexterous". In general numerical techniques have to be applied. A typical approach is finding the "differential solution" by formulating the Jacobian of the problem in the first step, and using some generalized inverse that selects a definite solution of the ambiguous set of the possible ones. Each of these solutions suffers from huge joint coordinate time-derivatives in the vicinity of the kinematic singularities that are evaded by "deforming" the original task. A quasi-differential approach was recently elaborated that, though calculates the Jacobian, instead "inverting" it, applies a fixed point iteration that automatically evades the singularities. However, its solution depends on the eigenvalues of the Jacobian, and it was found not "flexible enough" for all practical purposes. The Moore-Penrose solution can be generalized as an optimization task under constraints for quite complex cost functions. The hard constraint in this approach forces the exact solution of the inverse kinematic task when it is possible. Due to it the intent of "minimizing the costs" is "overridden" in the vicinity of the singularities. In the present paper, based on the formal structure of the Receding Horizon Controllers, an alternative solution is suggested that allows flexibility for relaxing the costs and the "hard constraints", too. Its computational complexity to some extent is reduced via evading the technique of gradient reduction. Simulation results are presented for a 7 DoF robot arm.

关键词： Inverse Kinematic Task Gradient Descent Method Reduced Gradient Method Moore-Penrose Pseudoinverse gram-schmidt algorithm Receding Horizon Control

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Direct Sequential Generation of Optimal Eigenvectors of the Symmetric Kernel of the Discrete Hankel Transform 67

Direct Sequential Generation of Optimal Eigenvectors of the ...

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67th IEEE International Midwest Symposium on Circuits and Systems (MWSCAS)

作者： Hanna, Magdy Tawfik Fayoum Univ Dept Engn Math & Phys Al Fayyum 63514 Egypt

ISBN: (纸本)9798350387186;9798350387179

Generating orthonormal Laguerre- Gaussian-power-like eigenvectors of the symmetric kernel matrix T of the discrete Hankel transform ( DHT) is the cornerstone in developing the discrete fractional Hankel transform (DFRHT). The direct sequential generation of the columns of each partition of the modal matrix of matrix T pertaining to one distinct eigenvalue is considered. The focus of the research is the assessment of the direct sequential algorithms: the gram-schmidt algorithm (GSA), modified gram-schmidt algorithm (MGSA), sequential orthogonal procrustes algorithm (SOPA) and direct sequential evaluation by constrained optimization algorithm (DSEOA). Although the four algorithms are theoretically equivalent, they are quite numerically distinct. The difference in performance becomes more pronounced as the order of the square matrix T becomes larger. The assessment shows that the MGSA outperforms both the GSA and the SOPA and that the DSEOA is by far the best in terms of computation accuracy.

关键词： discrete Hankel transform fractional transform modal matrix gram-schmidt algorithm modified gram-schmidt algorithm constrained minimization

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Robust Learning from Ortho-Diffusion Decompositions 16th

Robust Learning from Ortho-Diffusion Decompositions

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16th International Conference on Computer Analysis of Images and Patterns (CAIP)

作者： Gudivada, Sravan Bors, Adrian G. Univ York Dept Comp Sci York YO10 5GH N Yorkshire England

ISBN: (纸本)9783319231921;9783319231914

This paper describes a new classification method based on modeling data by embedding diffusions into orthonormal decompositions of graph-based data representations. The training data is represented by an adjacency matrix calculated using either the correlation or the covariance of the training set. The application of the modified gram-schmidt orthonormal decomposition alternating with diffusion and data reduction stages, is applied recursively at each scale level. The diffusion process is strengthening the representation pattern of representative features. Meanwhile, noise is removed together with non-essential detail during the data reduction stage. The proposed methodology is shown to be robust when applied to face recognition considering low image resolution and corruption by various types of noise.

关键词： Ortho-diffusion decompositions gram-schmidt algorithm Robust face recognition

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Speeding up the Reduced Gradient Method for Constrained Optimization 19

Speeding up the Reduced Gradient Method for Constrained Opti...

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19th IEEE World Symposium on Applied Machine Intelligence and Informatics (SAMI)

作者： Issa, Hazem Tar, Jozsef K. Obuda Univ Doctoral Sch Appl Informat & Appl Math Budapest Hungary Obuda Univ Res & Innovat Ctr Antal Bejczy Ctr Intelligent Robot Budapest Hungary

ISBN: (纸本)9781728180533

In various technical applications the local minimum of a differentiable cost function must be found under constraints that are interpreted as embedded hypersurfaces in the whole space of search. Generally Lagrange's "Reduced Gradient Method" can be applied for solving such problems in which the Lagrange multipliers associated with the individual constraint equations have important physical interpretation, therefore it is desirable to compute them. Though in special cases this algorithm can be replaced by closed form calculations via considering the "Auxiliary Function", in other cases the algorithmic realization cannot be avoided. In this paper it is shown that via avoiding the calculation of the individual Lagrange multipliers the algorithm can be made considerably faster especially if the constraint equations are appropriately handled. Simulation investigations are presented to substantiate the suggested method.

关键词： Reduced Gradient Auxiliary Function gram-schmidt algorithm Fixed Point Iteration Optimal Control

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ORTHO-DIFFUSION DECOMPOSITIONS FOR FACE RECOGNITION FROM LOW QUALITY IMAGES

ORTHO-DIFFUSION DECOMPOSITIONS FOR FACE RECOGNITION FROM LOW...

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IEEE International Conference on Image Processing (ICIP)

作者： Gudivada, Sravan Bors, Adrian G. Univ York Dept Comp Sci York YO10 5GH N Yorkshire England

ISBN: (纸本)9781479983391

We propose a new approach for recognizing human from images of low quality. An ortho-diffusion decomposition is used on graph representations of images. This is implemented by a recursive algorithm in three steps on either the covariance matrix or on the correlation of the training set. The first stage consists of an orthonormal decomposition implemented through the modified gram-schmidt with pivoting the columns. The other two stages consists of the data reduction and diffusion on graph representations. The data reduction ensures that the most significant features are preserved and together with the diffusion step ensures robustness to a variety of data corruption factors. The proposed methodology produces a set of ortho-diffusion bases representing the quintessential information from the training data set. The resulting orhto-diffusion bases are used to model face images when considering low resolution and corruption by various noise distributions.

关键词： gram-schmidt algorithm diffusion wavelets robust face recognition

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Cost Function-Free Optimization in Inverse Kinematics of Open Kinematic Chains 24th

Cost Function-Free Optimization in Inverse Kinematics of Ope...

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24th International Conference on Robotics in Alpe-Adria-Danube Region (RAAD)

作者： Tar, Jozsef K. Nadai, Laszlo Felde, Imre Rudas, Imre J. Obuda Univ Antal Bejczy Ctr Intelligent Robot Becsi Ut 96-B H-1034 Budapest Hungary Obuda Univ Smartlab Knowledge Ctr H-1034 Budapest Hungary

ISBN: (纸本)9783319212906;9783319212890

The traditional ways of solving various tasks "optimally" in control technology and robotics normally are based on the minimization of some cost function (or functional). On the basis of function minimization various "generalized inverse matrices" can be introduced that have special significance in the inverse kinematic tasks of redundant manipulators, where the possible solutions are ambiguous-therefore various choices are available. The solution suggested here tackles the question of optimality by the geometric interpretation of the simple and computationally efficient gram-schmidt algorithm. The method is presented via simulations using a redundant arm structure.

关键词： Open kinematic chain gram-schmidt algorithm Redundant robot arms Generalized inverse Inverse kinematic task

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