Introducing quaternion multi-valued neural networks with numerical examples

Greenblatt, Aaron; Agaian, Sos S.

doi:10.1016/j.ins.2017.09.057

Cited by 35 publications

(15 citation statements)

References 28 publications

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“…Because training an ELM is formulated as a least-squares problem, we provided tools for training general hypercomplexvalued ELM models using real linear algebra operations. Moreover, the realvalued formulation given by (31) depends almost solely on the multiplication table of the underlying hypercomplex algebra. Because its multiplication table uniquely determines a hypercomplex algebra, the definitions presented in this work allows for the implementation of ELMs in any hypercomplex algebra.…”

Section: Discussionmentioning

confidence: 99%

“…With regards to computational complexity, the hypercomplex-valued ELMs are much more time demanding than their real counterpart. The computational burden is mainly due to the transformations Φ L and Φ R required on ( 16), (19), and (31). For example, in our experiment, the training phase is carried out in 3.25s and 283.76s for the real and hypercomplex-valued models, respectively.…”

Section: Color Image Auto-encodingmentioning

confidence: 99%

“…These architectures are well adapted to multi-signal processing, meaning they can cope appropriately with phase information and rotations, for instance. Applications of HvNNs mainly revolve around signal and image processing [16][17][18][19][20][21][22][23][24][25], classification and prediction [26][27][28][29][30][31], and general image treatment such as auto-encoding and denoising [30,32].…”

Section: Introductionmentioning

confidence: 99%

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A General Framework for Hypercomplex-valued Extreme Learning Machines

Vieira,

Valle

2021

Preprint

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This paper aims to establish a framework for extreme learning machines (ELMs) on general hypercomplex algebras. Hypercomplex neural networks are machine learning models that feature higher-dimension numbers as parameters, inputs, and outputs. Firstly, we review broad hypercomplex algebras and show a framework to operate in these algebras through real-valued linear algebra operations in a robust manner. We proceed to explore a handful of well-known four-dimensional examples. Then, we propose the hypercomplex-valued ELMs and derive their learning using a hypercomplex-valued least-squares problem. Finally, we compare real and hypercomplex-valued ELM models' performance in an experiment on time-series prediction and another on color image autoencoding. The computational experiments highlight the excellent performance of hypercomplex-valued ELMs to treat high-dimensional data, including models based on unusual hypercomplex algebras.

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Section: Discussionmentioning

confidence: 99%

Section: Color Image Auto-encodingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A General Framework for Hypercomplex-valued Extreme Learning Machines

Vieira,

Valle

2021

Preprint

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“…On the other hand, since quaternion‐valued neural networks are superior to real‐valued and complex‐valued neural networks in processing high‐dimensional data and spatial transforms, in recent years, the study of quaternion‐valued neural networks has become a hot issue . However, due to the noncommutativity of the quaternion multiplication, the main method for studying the dynamical behavior of quaternion‐valued neural network systems is to decompose the considered neural network systems into real‐valued or complex‐valued systems …”

Section: Introductionmentioning

confidence: 99%

Existence and global exponential stability of almost periodic solution for quaternion‐valued high‐order Hopfield neural networks with delays via a direct method

Xiang

2020

Math Methods in App Sciences

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This paper deals with the existence and global exponential stability of almost periodic solutions for quaternion‐valued high‐order Hopfield neural networks with delays by a direct approach. Based on the contraction mapping principle, sufficient conditions are derived to ensure the existence and uniqueness of almost periodic solutions for the networks under consideration. By constructing a suitable Lyapunov function, the global exponential stability criterion of the almost periodic solution are derived. Finally, two numerical examples are given to illustrate the main results of this paper.

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“…When it comes to 3-D and 4-D data sources, such as measurements from seismometers, ultrasonic anemometers, and inertial body sensors, quaternions in quaternion domain H have inherent advantages over real vectors in representing 3-D and 4-D data owing to the natural ability to encode the cross-channel correlation and the accurate modeling of rotation and orientation [14]. In order to take advantage of the quaternion representation, a number of quaternionvalued neural models have been proposed, such as quaternion LMS algorithm [15], quaternion nonlinear adaptive filter [16], quaternion Kalman filter [17], quaternion independent component analysis algorithm [18], quaternion support vector machine [19], quaternion multi-valued neural networks [20], and quaternion ELM (QELM) [21]. Analogous with the complex case, the augmented quaternion statistics reveal that the covariance matrix is not adequate to capture the full second-order statistics of a quaternion vector [22], [23], which makes the Quaternion Widely Linear (QWL) processing become the optimal linear processing for general quaternion-valued signals.…”

Section: Introductionmentioning

confidence: 99%

Augmented Quaternion Extreme Learning Machine

Zhang

2019

IEEE Access

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As an efficient training strategy for single hidden layer neural networks, extreme learning machine, and its variants have been widely used due to its fast learning speed and superior generalization performance. However, when used for quaternion signal processing, extreme learning machine cannot make full use of the cross-channel correlation and quaternion statistics and thus often provides only suboptimal solutions. To this end, in this paper, we extend the extreme learning machine to quaternion domain and then propose two augmented quaternion extreme learning machine models for quaternion signal processing. These two models incorporate the involutions of the network input and hidden nodes, respectively, and can fully capture the second-order statistics of quaternion signals. In order to overcome the possible overfitting problem, two corresponding regularized augmented algorithms are also derived with the help of the generalized HR calculus. The superiority of the proposed models is verified by the simulation results. INDEX TERMS Extreme learning machine, quaternion signal processing, augmented quaternion statistics, regularization.

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Introducing quaternion multi-valued neural networks with numerical examples

Cited by 35 publications

References 28 publications

A General Framework for Hypercomplex-valued Extreme Learning Machines

A General Framework for Hypercomplex-valued Extreme Learning Machines

Existence and global exponential stability of almost periodic solution for quaternion‐valued high‐order Hopfield neural networks with delays via a direct method

Augmented Quaternion Extreme Learning Machine

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