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Deep Octonion Networks [article]

Jiasong Wu, Ling Xu, Youyong Kong, Lotfi Senhadji, Huazhong Shu
2019 arXiv   pre-print
This paper constructs a general framework of deep octonion networks (DONs) and provides the main building blocks of DONs such as octonion convolution, octonion batch normalization and octonion weight initialization  ...  In recent years, the deep complex networks (DCNs) and the deep quaternion networks (DQNs) have attracted more and more attentions.  ...  Octonion internal representation method The deep octonion networks are the octonion extension of the well-known deep real networks [25] , deep complex networks [65] and deep quaternion networks [67  ... 
arXiv:1903.08478v1 fatcat:whuogzxnizebfke3iz35heln3y

Hamiltonian and Q-inspired Neural Network-Based Machine Learning

Wieslaw Citko, Wieslaw Sienko
2020 IEEE Access  
It should be noted, however, that deep learning is currently driving a renaissance of interest in neural network research and applications (e.g., AI, big data, deep convolutional neural networks) [3]  ...  It is worth noting that as a potential direction for future deep learning research, geometric deep learning methods have been proclaimed [9] .  ... 
doi:10.1109/access.2020.3043035 fatcat:na3t4bmx4re2rcxfvwlog7n45m

A Parallel Algorithm for Dividing Octonions

Aleksandr Cariow, Janusz P. Paplinski
2021 Algorithms  
The article presents a parallel hardware-oriented algorithm designed to speed up the division of two octonions.  ...  In the synthesis of the discussed algorithm, the matrix representation of this operation was used, which allows us to present the division of octonions by means of a vector–matrix product.  ...  Klipkov for helpful clarifications regarding the dividing of octonions. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/a14110309 fatcat:ipi4dhljmfaajjtusu5wnhk4hi

Shifts in the Foundation: The Continual Modification and Generalization of Axioms and the Search for the Mathematical Principles that Underlie our Reality

Dr. Indrajit Patra , Et. al.
2021 Turkish Journal of Computer and Mathematics Education  
The study shall seek to explore the deep, underlying correspondence between the mathematical world of pure numbers and our physical reality.  ...  one-dimensional real numbers quantify many aspects of our day-to-day reality, complex numbers provide the mathematical foundations of quantum mechanics and also describe the behavior of more complicated quantum networks  ...  Professor Günaydin had through his researches sought to uncover some deep, underlying connections between String Theory, M-theory and supergravity and octonions themselves which parallels the attempts  ... 
doi:10.17762/turcomat.v12i2.1126 fatcat:q6deypu3ibaaxof7b57snywbuu

Table of contents

2020 IEEE Transactions on Neural Networks and Learning Systems  
Zhang Metacognitive Octonion-Valued Neural Networks as They Relate to Time Series Analysis ............................... ..............................................................................  ...  Fu Distribution-Free Probability Density Forecast Through Deep Neural Networks ............................................ .............................................................................  ... 
doi:10.1109/tnnls.2020.2968822 fatcat:kjzhsdgiqneyblde3226ko56hi

Quaternion Knowledge Graph Embeddings [article]

Shuai Zhang and Yi Tay and Lina Yao and Qi Liu
2019 arXiv   pre-print
More recently, convolution neural networks [Dettmers et al., 2018] , graph convolutional networks [Schlichtkrull et al., 2018] , and deep memory networks [Wang et al., 2018] also show promising performance  ...  Knowledge graph embedding with entity neighbors and deep memory network. arXiv preprint arXiv:1808.03752, 2018. Zhen Wang, Jianwen Zhang, Jianlin Feng, and Zheng Chen.  ... 
arXiv:1904.10281v3 fatcat:a2wr2eycbrb7dhbmzmsfu4xgcu

Hypercomplex-Valued Recurrent Correlation Neural Networks [article]

Marcos Eduardo Valle, Rodolfo Anibal Lobo
2020 arXiv   pre-print
Recurrent correlation neural networks (RCNNs), introduced by Chiueh and Goodman as an improved version of the bipolar correlation-based Hopfield neural network, can be used to implement high-capacity associative  ...  Examples with bipolar, complex, hyperbolic, quaternion, and octonion-valued RCNNs are given to illustrate the theoretical results.  ...  Furthermore, RCNNs are closely related to the dense associative memory model introduced by Krotov and Hopfield to establish the duality between associative memories and deep learning [56, 57] .  ... 
arXiv:2002.00027v1 fatcat:dziplju4gjgd7e3e6koupfqire

Braided fermions from Hurwitz algebras [article]

Niels G Gresnigt
2018 arXiv   pre-print
That is, the ribbon spectrum can be related to octonion algebras.  ...  These braided fermion states can be identified with the basis states of the minimal left ideals of the Clifford algebra Cℓ(6), generated from the nested left actions of the complex octonions C⊗O on itself  ...  This is not the only hint at a deep connection between the two models.  ... 
arXiv:1901.01312v1 fatcat:hqdks2yvbjhk3fwhuou25jqvgu

Multi-Source and Multi-Temporal Image Fusion on Hypercomplex Bases

Andreas Schmitt, Anna Wendleder, Rüdiger Kleynmans, Maximilian Hell, Achim Roth, Stefan Hinz
2020 Remote Sensing  
This study combined Sentinel-1 and Sentinel-2 on an Octonion basis as well as Sentinel-2 and ALOS-PALSAR-2 on a Sedenion basis.  ...  The approach is based on artificial neural networks like Convolutional Neural Networks (CNN) [57] [58] [59] , Convolutional Sparse Coding [57, 60] , or Stacked Autoencoders [61] .  ...  The usage of a neural network classification method still promises 49 %.  ... 
doi:10.3390/rs12060943 fatcat:kyzgo56ynncb5ccz2v5r5qu5yq

Matroid Theory and Supergravity [article]

J. A. Nieto
1998 arXiv   pre-print
We show a connection between this last theorem and octonions which at the same time, as it is known, are related to the Englert's solution of D = 11 supergravity.  ...  that the relation between these three concepts must have a deep significate in nature.  ...  Concrete applications may be found in circuit theory, network-flow theory, linear and integer programming and the theory (01)-matrices, for example.  ... 
arXiv:hep-th/9807107v1 fatcat:rnzpina42bdunp432xcucirqfu

Placeholder Substructures I: The Road from NKS to Scale-Free Networks Is Paved with Zero-Divisors [article]

Robert P. C. de Marrais
2007 arXiv   pre-print
derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks  ...  excludes a different Octonion from its set.  ...  Another type of 6-cycle lanyard, the 2-Sail, 5-Assessor "Bow-Tie" whose doubly passed-through "knot" must be in the Zigzag, has deep symmetry-breaking properties that may well play a fundamental role in  ... 
arXiv:math/0703745v3 fatcat:ncwzj7xjorbqrgvlj4fsc6kfmq

Convolutional Hypercomplex Embeddings for Link Prediction [article]

Caglar Demir, Diego Moussallem, Stefan Heindorf, Axel-Cyrille Ngonga Ngomo
2021 arXiv   pre-print
QMult and OMult can be considered as quaternion and octonion extensions of previous state-of-the-art approaches, including DistMult and ComplEx.  ...  The residual learning framework facilitates the training of deep neural networks.  ...  However, the octonion multiplication in neural networks and learning octonion-valued knowledge graph embeddings had not yet been fully explored. Convolutional Hypercomplex Embeddings Motivation.  ... 
arXiv:2106.15230v2 fatcat:phiv2jebo5h2zb7uflt4r6l5lu

Effecting Semantic Network Bricolage via Infinite-Dimensional Zero-Divisor Ensembles

Robert P. C. de Marrais, Thothic Technology Partners
2009 Complex Systems  
Continuing arguments presented [1] or announced [2, 3] here, zerodivisor (ZD) foundations for scale-free networks (evinced, in particular, in the "fractality" of the Internet) are decentralized.  ...  Any HBK houses, in a cowbird's nest, exactly one copy of the (ZD-free) octonions, the recursive basis for all ZD ensembles.  ...  Effecting Semantic Network Bricolage Lemma 2.  ... 
doi:10.25088/complexsystems.18.1.75 fatcat:wntd7wg4vjghxbeaumgvkij2uu

Voyage by Catamaran: Effecting Semantic Network "Bricolage" via Infinite-Dimensional Zero-Divisor Ensembles [article]

Robert P. C. de Marrais
2008 arXiv   pre-print
Any HBK houses, in a "cowbird's nest," exactly one copy of the (ZD-free) octonions, the recursive basis for all ZD ensembles.  ...  Continuing arguments presented [1] or announced [2][3] in "Complex Systems," zero-divisor (ZD) foundations for "scale-free" networks (evinced, in particular, in the "fractality" of the Internet) are decentralized  ...  We have, then, a pure octonion copy.  ... 
arXiv:0804.3416v2 fatcat:zxixb24p2ffxnakpks56n64mlq

Generalizing Complex/Hyper-complex Convolutions to Vector Map Convolutions [article]

Chase J Gaudet, Anthony S Maida
2020 arXiv   pre-print
We show that the core reasons that complex and hypercomplex valued neural networks offer improvements over their real-valued counterparts is the weight sharing mechanism and treating multidimensional data  ...  We perform three experiments to show that these novel vector map convolutions seem to capture all the benefits of complex and hyper-complex networks, such as their ability to capture internal latent relations  ...  The weight initialization for vector map networks uses the same procedure seen in both deep complex networks [25] and deep quaternion networks [5] .  ... 
arXiv:2009.04083v1 fatcat:g6ewtuvjancehiznhphcnow3ki
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