The Experts below are selected from a list of 300 Experts worldwide ranked by ideXlab platform
Jenna Reis - One of the best experts on this subject based on the ideXlab platform.
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the spectral Connection Matrix for any change of basis within the classical real orthogonal polynomials
Mathematics, 2015Co-Authors: T Bella, Jenna ReisAbstract:The Connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the Connection problem may be solved by directly computing the change–of–basis Matrix, this approach is computationally expensive. A recent approach to solving the Connection problem involves the use of the spectral Connection Matrix, which is a Matrix whose eigenvector Matrix is the desired change–of–basis Matrix. In Bella and Reis (2014), it is shown that for the Connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral Connection Matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral Connection Matrix for Connections between real orthogonal polynomial families. In particular, for the Connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral Connection Matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral Connection Matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral Connection Matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).
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The spectral Connection Matrix for classical real orthogonal polynomials
2015Co-Authors: Jenna ReisAbstract:Sets of orthogonal polynomials are bases for polynomial spaces Pn. As a result, polynomials can be expressed in coefficients relative to a particular family of orthogonal polynomials. The Connection problem refers to the task of converting from coefficients in one of these bases to coefficients in another. The entries of the Matrix that applies such a change of basis, known as the Connection coefficients, are well-known values that can be computed via direct computation or Matrix inversion; however this can be computationally expensive. Thus their accurate and efficient computation is a relevant topic of research in numerical linear algebra, and can be found in the current literature. The two manuscripts included in this thesis address the Connection problem. In the first manuscript, a Connection within the classical real orthogonal polynomials of a single parameter (Hermite, Laguerre,and Gegenbauer) is discussed. The spectral Connection Matrix related to a Connection Matrix is defined. It is also shown that this spectral Connection Matrix in each case within the singleparameter classical families is quasiseparable, with specific generators provided. Additionally this manuscript proposes an algorithm that efficiently computes the desired Connection Matrix given the generators of its corresponding spectral Connection Matrix. The second manuscript dramatically generalizes the result of the first. It addresses the structure of the spectral Connection Matrix associated with a much broader group of Connections. The target family is allowed to be any of the classical types, including Jacobi. The source family is allowed to be any of the classical types or Bessel, which is not considered classical here. In these cases it is shown that once again the spectral Connection Matrix is quasiseparable, and specific generators are provided. The algorithm from the first manuscript allows for the efficient computation of the desired Connection Matrix given the generators of the associated spectral Connection Matrix. The appendix at the conclusion provides some details for the reader’s reference. It begins with a review of orthogonal polynomials, and highlights the classical types. It then provides a review of some basic linear algebra concepts that are relevant to the manuscripts, and concludes with a survey of quasiseparable matrices. The appendix also references research activity in the field.
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the spectral Connection Matrix for classical orthogonal polynomials of a single parameter
Linear Algebra and its Applications, 2014Co-Authors: T Bella, Jenna ReisAbstract:Abstract In this paper we study the so-called Connection problem of, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. We restrict our current study to the classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer (including Legendre) families. The computational tool for this work is the class of quasiseparable matrices. While the relationships between orthogonal polynomials and rank-structured matrices such as quasiseparable matrices are very well-known, in this paper we investigate a more recently considered relationship. We prove that, while the Matrix that implements the desired Connection is not itself quasiseparable, it is an eigenvector Matrix of one that is quasiseparable. We suggest to refer to this structured Matrix as the spectral Connection Matrix. Finally, we present a simple algorithm exploiting the computationally favorable properties of quasiseparable matrices to implement the desired change of basis. By exploiting the quasiseparable structure, this algorithm enjoys an order of magnitude reduction of complexity as compared to the simple method of inverting the Connection Matrix directly. While not the focus of the paper, some very preliminary numerical experimentation shows some positive indications that even with this reduction in complexity the accuracy of the resulting change of basis algorithm is comparable to that of inverting the Connection Matrix directly.
Toshimichi Saito - One of the best experts on this subject based on the ideXlab platform.
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Connection sparsity versus orbit stability in dynamic binary neural networks
2017 International Joint Conference on Neural Networks (IJCNN), 2017Co-Authors: Ryuji Sato, Shunsuke Aoki, Toshimichi SaitoAbstract:This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network Connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary Connection Matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the Connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the Connection Matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.
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IJCNN - Connection sparsity versus orbit stability in dynamic binary neural networks
2017 International Joint Conference on Neural Networks (IJCNN), 2017Co-Authors: Ryuji Sato, Shunsuke Aoki, Toshimichi SaitoAbstract:This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network Connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary Connection Matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the Connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the Connection Matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.
Chia-lun John Hu - One of the best experts on this subject based on the ideXlab platform.
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Noniterative learning in perceptrons implemented by an ultrafast-learning character-recognition scheme
Proceedings of International Conference on Neural Networks (ICNN'96), 1996Co-Authors: Chia-lun John HuAbstract:As we studied in the last five years, for an artificial perceptron consisting of hard-limited neurons, the Connection Matrix to meet a given input-output mapping can actually be obtained noniteratively in one step if the given mapping satisfies a certain PLI condition. Whenever the given mapping satisfies this condition, generally there exists infinitively many solutions for the Connection Matrix. One can then select an optimum solution such that in the recognition mode, the recognition of any untrained input vectors becomes optimally robust. The "learning" here (or the obtaining of the Connection Matrix from the given mapping) should be very fast because the learning process is noniterative and one-step. The recognition of untrained inputs here should be optimally robust because the optimum analysis here is independent of the learning method we use. This paper reports the theoretical analysis of this noniterative learning scheme and the design and the experiment of a practical ultrafast-learning, character-recognition scheme derived from this theory.
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ICNN - Noniterative learning in perceptrons implemented by an ultrafast-learning character-recognition scheme
Proceedings of International Conference on Neural Networks (ICNN'96), 1996Co-Authors: Chia-lun John HuAbstract:As we studied in the last five years, for an artificial perceptron consisting of hard-limited neurons, the Connection Matrix to meet a given input-output mapping can actually be obtained noniteratively in one step if the given mapping satisfies a certain PLI condition. Whenever the given mapping satisfies this condition, generally there exists infinitively many solutions for the Connection Matrix. One can then select an optimum solution such that in the recognition mode, the recognition of any untrained input vectors becomes optimally robust. The "learning" here (or the obtaining of the Connection Matrix from the given mapping) should be very fast because the learning process is noniterative and one-step. The recognition of untrained inputs here should be optimally robust because the optimum analysis here is independent of the learning method we use. This paper reports the theoretical analysis of this noniterative learning scheme and the design and the experiment of a practical ultrafast-learning, character-recognition scheme derived from this theory.
Ryuji Sato - One of the best experts on this subject based on the ideXlab platform.
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Connection sparsity versus orbit stability in dynamic binary neural networks
2017 International Joint Conference on Neural Networks (IJCNN), 2017Co-Authors: Ryuji Sato, Shunsuke Aoki, Toshimichi SaitoAbstract:This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network Connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary Connection Matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the Connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the Connection Matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.
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IJCNN - Connection sparsity versus orbit stability in dynamic binary neural networks
2017 International Joint Conference on Neural Networks (IJCNN), 2017Co-Authors: Ryuji Sato, Shunsuke Aoki, Toshimichi SaitoAbstract:This paper considers two basic problems in artificial neural networks that can generate various binary periodic orbits. The first problem is relation between sparsity of network Connection and stability of a target periodic orbit. The second problem is comparison between digital circuits and artificial neural networks in the orbit stability. We consider these problems in dynamic binary neural networks characterized by the signum activation function and ternary Connection Matrix. Performing basic numerical experiments, we give conjectures for the two problems. First, as the Connection sparsity increases, the orbit stability varies. There exists suitable sparsity in which the orbit stability is very strong. Second, as the Connection Matrix approaches to the most sparse case, the dynamic binary neural network approaches to an equivalent system to the shift register that has no stable periodic orbit.
A. Melchinger - One of the best experts on this subject based on the ideXlab platform.
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Calibration of a hand/eye Matrix and a Connection Matrix using relative pose measurements
IEEE Transactions on Systems Man and Cybernetics - Part A: Systems and Humans, 1998Co-Authors: H. Zhuang, A. MelchingerAbstract:The problem investigated in this paper is an extension of the robotic hand/eye calibration problem. The system consists of a robot, a gimbal, and a three-dimensional (3D) sensor. It is assumed that the sensor, the gimbal, and the robot are calibrated in advance; therefore, their inaccuracy is negligible. The task is to determine the hand/eye Matrix that relates the sensor coordinate frame to the first gimbal coordinate frame and the Connection Matrix that relates the last gimbal coordinate frame to the robot tool frame. The paper focuses on a special case of this problem. The robot is an x-y-z type and the gimbal has two rotary joints. Linear and iterative methods are presented, along with the discussion on issues such as solution uniqueness and parameter observability. Simulation and experimental results are presented to demonstrate the feasibility of the method.
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Calibration of a hand/eye Matrix and a Connection Matrix using relative pose measurements
Proceedings of International Conference on Robotics and Automation, 1997Co-Authors: H. Zhuang, A. MelchingerAbstract:The problem investigated in this paper is an extension of the robotic hand/eye calibration problem. The system consists of a robot, a gimbal and a three dimensional sensor. The task is to determine the hand/eye Matrix that relates the sensor coordinate frame to the first coordinate frame in the gimbal and the Connection Matrix that relates the first coordinate frame in the gimbal to the tool frame in the robot. The paper focuses on a special case of the mentioned problem. In this study, the robot is a x-y-z type and the gimbal has two rotary joints. Two solution methods, one of which is linear and another is nonlinear, are presented in this paper. Issues such as the uniqueness of the linear solution and the observability of error parameters are also investigated. Simulation and experimental results are given to demonstrate the feasibility of the method.
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ICRA - Calibration of a hand/eye Matrix and a Connection Matrix using relative pose measurements
Proceedings of International Conference on Robotics and Automation, 1997Co-Authors: H. Zhuang, A. MelchingerAbstract:The problem investigated in this paper is an extension of the robotic hand/eye calibration problem. The system consists of a robot, a gimbal and a three dimensional sensor. The task is to determine the hand/eye Matrix that relates the sensor coordinate frame to the first coordinate frame in the gimbal and the Connection Matrix that relates the first coordinate frame in the gimbal to the tool frame in the robot. The paper focuses on a special case of the mentioned problem. In this study, the robot is a x-y-z type and the gimbal has two rotary joints. Two solution methods, one of which is linear and another is nonlinear, are presented in this paper. Issues such as the uniqueness of the linear solution and the observability of error parameters are also investigated. Simulation and experimental results are given to demonstrate the feasibility of the method.