A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called $\mathcal{B}$-projection functions. Broadly speaking, a $\mathcal{B}$-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras

Neural Computing in Quaternion Algebra

兵庫県立大学201

Deep Quaternion Networks

The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hyper-complex numbers, quaternions specifically, and provide the architecture components needed to build deep quaternion networks. We develop the theoretical basis by reviewing quaternion convolutions, developing a novel quaternion weight initialization scheme, and developing novel algorithms for quaternion batch-normalization. These pieces are tested in a classification model by end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation model by end-to-end training on the KITTI Road Segmentation data set. These quaternion networks show improved convergence compared to real-valued and complex-valued networks, especially on the segmentation task, while having fewer parametersComment: IJCNN 2018, 8 pages, 1 figur

Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work

Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work

Selected aspects of complex, hypercomplex and fuzzy neural networks

This short report reviews the current state of the research and methodology on theoretical and practical aspects of Artificial Neural Networks (ANN). It was prepared to gather state-of-the-art knowledge needed to construct complex, hypercomplex and fuzzy neural networks. The report reflects the individual interests of the authors and, by now means, cannot be treated as a comprehensive review of the ANN discipline. Considering the fast development of this field, it is currently impossible to do a detailed review of a considerable number of pages. The report is an outcome of the Project 'The Strategic Research Partnership for the mathematical aspects of complex, hypercomplex and fuzzy neural networks' meeting at the University of Warmia and Mazury in Olsztyn, Poland, organized in September 2022.Comment: 46 page

A quaternion deterministic monogenic CNN layer for contrast invariance

Deep learning (DL) is attracting considerable interest as it currently achieves remarkable performance in many branches of science and technology. However, current DL cannot guarantee capabilities of the mammalian visual systems such as lighting changes. This paper proposes a deterministic entry layer capable of classifying images even with low-contrast conditions. We achieve this through an improved version of the quaternion monogenic wavelets. We have simulated the atmospheric degradation of the CIFAR-10 and the Dogs and Cats datasets to generate realistic contrast degradations of the images. The most important result is that the accuracy gained by using our layer is substantially more robust to illumination changes than nets without such a layer.The authors would like to thank to CONACYT and Barcelona supercomputing Center. Sebastián Salazar-Colores (CVU 477758) would like to thank CONACYT (Consejo Nacional de Ciencia y Tecnología) for the financial support of his PhD studies under Scholarship 285651. Ulises Moya and Ulises Cortés are member of the Sistema Nacional de Investigadores CONACyT.Peer ReviewedPostprint (author's final draft