Abstract
This paper presents a new algorithm for multiplying two Kaluza numbers. Performing this operation directly requires 1024 real multiplications and 992 real additions. We presented in a previous paper an effective algorithm that can compute the same result with only 512 real multiplications and 576 real additions. More effective solutions have not yet been proposed. Nevertheless, it turned out that an even more interesting solution could be found that would further reduce the computational complexity of this operation. In this article, we propose a new algorithm that allows one to calculate the product of two Kaluza numbers using only 192 multiplications and 384 additions of real numbers.
Highlights
The permanent development of the theory and practice of data processing, as well as the need to solve increasingly complex problems of computational intelligence, inspire the use of complex and advanced mathematical methods and formalisms to represent and process big multidimensional data arrays
We presented a new effective algorithm for calculating the product of two Kaluza numbers
The use of this algorithm reduces the computational complexity of multiplications of Kaluza numbers, reducing implementation complexity and leading to a high-speed resource-effective architecture suitable for parallel implementation on VLSI
Summary
The permanent development of the theory and practice of data processing, as well as the need to solve increasingly complex problems of computational intelligence, inspire the use of complex and advanced mathematical methods and formalisms to represent and process big multidimensional data arrays. Hypercomplex numbers [1] are used in various fields of data processing, including digital signal and image processing, machine graphics, telecommunications, and cryptography [2,3,4,5,6,7,8,9,10]. Their use in brain-inspired computation and neural networks has been largely limited due to the lack of comprehensive and all-inclusive information processing and deep learning techniques. The object of our research was hypercomplex-valued convolutional neural networks using 32-dimensional Kaluza numbers
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