Algebraic foundations of split hypercomplex nonlinear adaptive filtering

Hitzer, Eckhard

doi:10.1002/mma.2660

Cited by 29 publications

(12 citation statements)

References 25 publications

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“…Convergence analysis of the quaternion neural network is important and several studies have been conducted. (7,13,16) In this study, the difference in the cost function ∆J (k) is derived as…”

Section: Self-tuning Controllermentioning

confidence: 99%

Design of Quaternion-Neural-Network-Based Self-Tuning Control Systems

2017

Sensors and Materials

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In this study, we investigate the control performance of an adaptive controller using a multilayer quaternion neural network. The control system is a self-tuning controller, the control parameters of which are tuned online by the quaternion neural network to track plant output to follow the desired output generated by a reference model. A proportional-integral-derivative (PID) controller is used as a conventional controller, the parameters of which are tuned by the quaternion neural network. Computational experiments to control a single-input single-output (SISO) discrete-time nonlinear plant are conducted to evaluate the capability and characteristics of the quaternion-neural-networkbased self-tuning PID controller. Experimental results show the feasibility and effectiveness of the proposed controller.

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Section: Self-tuning Controllermentioning

confidence: 99%

Design of Quaternion-Neural-Network-Based Self-Tuning Control Systems

2017

Sensors and Materials

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“…Hypercomplex AFs available in the literature make use of quaternion algebra [24], [25] and even GA theory [26]. However, the error vector therein has the form e = y − rx, which is not appropriate to model rotation error since it lacks r multiplying x from the right.…”

Section: B the Estimation Problem In Gamentioning

confidence: 99%

Geometric-Algebra LMS Adaptive Filter and Its Application to Rotation Estimation

Lopes

Al-Nuaimi²,

Lopes

2016

IEEE Signal Process. Lett.

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This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D pointclouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.

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“…Kobayashi and Nakajima proposed twisted quaternionic neural networks and demonstrated by computer simulations that they tended to avoid plateaus and local minima [16]. Quaternions have been also applied to signal processing [17][18][19]. Moreover, several hyperbolic neural networks have been proposed.…”

Section: Introductionmentioning

confidence: 99%

Hyperbolic Hopfield neural networks with four‐state neurons

Kobayashi

2016

IEEJ Transactions Elec Engng

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In recent years, applications of neural networks with Clifford algebra have become widespread. Clifford algebra is also referred to as geometric algebra and is useful in dealing with geometric objects. Hopfield neural networks with Clifford algebra, such as complex numbers and quaternions, have been proposed. However, it has been difficult to construct Hopfield neural networks by Clifford algebra with positive part of the signature, such as hyperbolic numbers. Hyperbolic numbers are useful algebra to deal with hyperbolic geometry. Kuroe proposed hyperbolic Hopfield neural networks and provided their continuous activation functions and stability conditions. However, the learning algorithm has not been provided. In this paper, we provide two quantized activation functions and the primitive learning algorithm satisfying the stability condition. We also perform computer simulations and compare the activation functions.

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Algebraic foundations of split hypercomplex nonlinear adaptive filtering

Cited by 29 publications

References 25 publications

Design of Quaternion-Neural-Network-Based Self-Tuning Control Systems

Design of Quaternion-Neural-Network-Based Self-Tuning Control Systems

Geometric-Algebra LMS Adaptive Filter and Its Application to Rotation Estimation

Hyperbolic Hopfield neural networks with four‐state neurons

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