Polarized Signal Classification by Complex and Quaternionic Multi-Layer Perceptrons

Buchholz, Sven; Bihan, Nicolas Le

doi:10.1142/s0129065708001403

Cited by 64 publications

(30 citation statements)

References 13 publications

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“…In recent years, quaternion algebra has been successfully applied to numerous problems in physics [25], computer graphics [1], signal processing and communications [5,9,11,17,23,24,26,29]. In these applications, quaternions have allowed for a reduction in the number of parameters and operations involved.…”

Section: Introductionmentioning

confidence: 99%

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Xia

Mandic

2016

IEEE Trans. Neural Netw. Learning Syst.

112

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The optimization of real scalar functions of quaternion variables, such as the mean square error or array output power, underpins many practical applications. Solutions often require the calculation of the gradient and Hessian, however, real functions of quaternion variables are essentially non-analytic. To address this issue, we propose new definitions of quaternion gradient and Hessian, based on the novel generalized HR (GHR) calculus, thus making possible efficient derivation of optimization algorithms directly in the quaternion field, rather than transforming the problem to the real domain, as is current practice. In addition, unlike the existing quaternion gradients, the GHR calculus allows for the product and chain rule, and for a one-to-one correspondence of the proposed quaternion gradient and Hessian with their real counterparts. Properties of the quaternion gradient and Hessian relevant to numerical applications are elaborated, and the results illuminate the usefulness of the GHR calculus in greatly simplifying the derivation of the quaternion least mean squares, and in quaternion least square and Newton algorithm. The proposed gradient and Hessian are also shown to enable the same generic forms as the corresponding real-and complex-valued algorithms, further illustrating the advantages in algorithm design and evaluation.

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Section: Introductionmentioning

confidence: 99%

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Xia

Mandic

2016

IEEE Trans. Neural Netw. Learning Syst.

112

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“…A comprehensive comparison of the performances is provided between the training algorithm for the feedforward QMLP [8], [38] and the nonlinear finite impulse response filters trained with the QMLP learning algorithm (QMLP-FIR) [11], adaptive amplitude split quaternion adaptive filtering algorithm (AASQAFA) [11], real-valued nonlinear gradient descent (NGD) [37], and the proposed algorithms based on fully quaternion nonlinear functions, i.e., QNGD and AQNGD. The quaternion multilayer perceptron-finite impulse response (QMLP-FIR), AASQAFA, NGD, QNGD, and AQNGD were implemented with a filter length L, whereas the QMLP had one hidden layer comprising of L input neurons, three hidden neurons, and one output neuron.…”

Section: Simulationsmentioning

confidence: 99%

Quaternion-Valued Nonlinear Adaptive Filtering

Ujang

Took

Mandic

2011

IEEE Trans. Neural Netw.

184

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Abstract-A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real-and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.Index Terms-Augmented quaternion statistics, H-circularity, nonlinear adaptive filtering, quaternion least mean square, widely linear modeling, widely linear quaternion least mean square, wind prediction.

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“…They have received much attention in recent years. For example, quaternion signal processing has encountered applications in wind forecasting [1], aerospace [2], computer graphics problems [3], image processing [4], vector sensor [5], processing of polarized waves [6], and design of space-time block codes [7].…”

Section: Introductionmentioning

confidence: 99%

Semi-widely linear estimation of Cη-proper quaternion random signal vectors under Gaussian and stationary conditions

Navarro-Moreno

Ruiz-Molina

2016

Signal Processing

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Polarized Signal Classification by Complex and Quaternionic Multi-Layer Perceptrons

Cited by 64 publications

References 13 publications

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Optimization in Quaternion Dynamic Systems: Gradient, Hessian, and Learning Algorithms

Quaternion-Valued Nonlinear Adaptive Filtering

Semi-widely linear estimation of Cη-proper quaternion random signal vectors under Gaussian and stationary conditions

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