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Introduction
Publications
Publications (37)
The brief presents the results of synthesizing efficient algorithms for implementing the basic data-processing macro operations used in tessarine-valued neural networks. These macro operations primarily include the macro operation of multiplication of two tessarines: the macro operation of calculating the inner product of two tessarine-valued vecto...
This brief presents the results of a study of the possibilities of reducing the arithmetic complexity of computing basic operations in octonionic neural networks and also proposes new algorithmic solutions for efficiently performing these operations. Here, we primarily mean the operation of multiplying octonions, the operation of computing the dot...
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 ye...
In this paper, we present several resource-efficient algorithmic solutions regarding the fully parallel hardware implementation of the basic filtering operation performed in the convolutional layers of convolution neural networks. In fact, these basic operations calculate two inner products of neighboring vectors formed by a sliding time window fro...
In this work, a rationalized algorithm for calculating the quotient of two quaternions is presented which reduces the number of underlying real multiplications. Hardware for fast multiplication is much more expensive than hardware for fast addition. Therefore, reducing the number of multiplications in VLSI processor design is usually a desirable ta...
In this paper, we present several resource-efficient algorithmic solutions regarding the fully parallel hardware implementation of the basic filtering operation performed in the convolutional layers of convolution neural networks. In fact, these basic operations calculate two inner products of neighboring vectors formed by a sliding time window fro...
In this article, we analyze algorithmic ways to reduce the arithmetic complexity of calculating quaternion-valued linear convolution and also synthesize a new algorithm for calculating this convolution. During the synthesis of the discussed algorithm, we use the fact that quaternion multiplication may be represented as a matrix-vector product. The...
This paper proposes a new fast algorithm for calculating the discrete fractional Hadamard transform for data vectors whose size N is a power of two. A direct method for the calculation of the discrete fractional Hadamard transform requires O(N
<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup>
) m...
In this paper, we have proposed a novel VLSI-oriented parallel algorithm for quaternion-based rotation in 3D space. The advantage of our algorithm is a reduction the number of multiplications through replacing part of them by less costly squarings. The algorithm uses Logan’s trick, which proposes to replace the calculation of the product of two num...
This paper presents a structural design of the hardware-efficient module for implementation of convolution neural network (CNN) basic operation with reduced implementation complexity. For this purpose we utilize some modification of the Winograd minimal filtering method as well as computation vectorization principles. This module calculate inner pr...
In this work, a fast algorithm for quaternion-based 4D rotation is presented which reduces the number of underlying real multiplications. Performing a quaternion-based rotation using rotation matrix takes 32 multiplications and 60 additions of real numbers while the proposed algorithm can compute the same result in only 16 real multiplications (or...
In this chapter an algorithm for computing the Vandermonde matrix-vector product is presented. The main idea of constructing this algorithm is based on the using of Winograd’s formula for inner product computation. Multiplicative complexity of the proposed algorithm is less than multiplicative complexity of the schoolbook (naïve) method of calculat...
In this paper, new schemes for a squarer, multiplier and divider of complex numbers are proposed. Traditional structural solutions for each of these operations require the presence some number of general-purpose binary multipliers. The advantage of our solutions is a removing of multiplications through replacing them by less costly squarers. We use...
In this paper, we offer and discuss three efficient structural solutions for the hardware-oriented implementation of discrete quaternion Fourier transform basic operations with reduced implementation complexities. The first solution: a scheme for calculating sq product, the second solution: a scheme for calculating qt product, and the third solutio...
In this paper, we have proposed a novel VLSI-oriented approach to computing the rotation matrix entries from the quaternion coefficients. The advantage of this approach is the complete elimination of multiplications and replacing them by less costly squarings. Our approach uses Logan's identity, which proposes to replace the calculation of the prod...
In this paper we introduce efficient algorithm for the multiplication of biquaternions. The direct multiplication of two biquaternions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the Pauli numbers with 24 real multiplications and 64 real additions. During s...
This paper presents the derivation of a new algorithm for multiplying of two
Kaluza numbers. Performing this operation directly requires 1024 real
multiplications and 992 real additions. The proposed algorithm can compute the
same result with only 512 real multiplications and 576 real additions. The
derivation of our algorithm is based on utilizing...
We present an efficient algorithm to multiply two arbitrary biquaternions. The schoolbook multiplication of two biquaternions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of the biquaternions with 24 real multiplications and 56 real additions. During synthesis...
In this paper we introduce an efficient algorithm for the multiplication of biquaternions. The direct multiplication of two biquaternions requires 64 real multiplications and 56 real additions. More effective solutions still do not exist. We show how to compute a product of biquaternions with 24 real multiplications and 64 real additions. During sy...
In this paper we introduce efficient algorithm for the multiplication of
split-octonions. The direct multiplication of two split-octonions requires 64
real multiplications and 56 real additions. More effective solutions still do
not exist. We show how to compute a product of the split-octonions with 28 real
multiplications and 92 real additions. Du...
We present an efficient algorithm to multiply two hyperbolic octonions. The
direct multiplication of two hyperbolic octonions requires 64 real
multiplications and 56 real additions. More effective solutions still do not
exist. We show how to compute a product of the hyperbolic octonions with 26
real multiplications and 92 real additions. During syn...
In this work a rationalized algorithm for Dirac numbers multiplication is
presented. This algorithm has a low computational complexity feature and is
well suited to FPGA implementation. The computation of two Dirac numbers
product using the na\"ive method takes 256 real multiplications and 240 real
additions, while the proposed algorithm can comput...
In this paper we present a hardware-oriented algorithm for constant
matrix-vector product calculating, when the all elements of vector and matrix
are complex numbers. The proposed algorithm versus the naive method of
analogous calculations drastically reduces the number of multipliers required
for FPGA implementation of complex-valued constant matr...
In this paper we introduce efficient algorithm for the multiplication of trigintaduonions. The direct multiplication of two trigintaduonions requires 1024 real multiplications and 992 real additions. We show how to compute a trigintaduonion product with 498 real multiplications and 943 real additions. During synthesis of the discussed algorithm we...
In this paper two different approaches to the rationalization of FDWT and IDWT basic operations execution with the reduced number of multiplications are considered. With regard to the well-known approaches, the direct implementation of the above operations requires 2L multiplications for the execution of FDWT and IDWT basic operation plus 2(L-1) ad...
In this paper we introduce an efficient algorithm for the multiplication of Pauli numbers. The direct multiplication of two Pauli numbers requires 64 real multiplications and 56 real additions. More effective
solutions still do not exist. We show how to compute a product of the Pauli numbers with 24 conventional multiplications, 8 multiplications b...
We propose an original algorithmic solution for multiplication of octonions. In previously published algorithms for computing the product of octonions the number of multiplications has been reduced by significantly increasing number of additions and shifts. A dignity of the proposed solutions is to reduce by 25% the number of multiplications needed...
This paper presents a high-speed parallel 3x3 matrix multiplier structure. To reduce the hardware complexity of the multiplier structure, we propose to modify the Makarov’s algorithm for 3x3 by 3x3 matrix multiplication The process of matrix product calculation is successively decomposed so that a minimal set of multipliers and fewer adders are use...
In paper a rationalized approaches to Discrete Wavelet Transform (DWT) co-efficients computing, based on modified algorithm of DWT base operation execution, with less multiplication operations are presented. Those several intuitive ways allow to decrease computational complexity of hardware operational units for DWT basic operation computation and...
In this note we present the algorithm for vector-matrix product calculating for vectors and matrices whose elements are complex numbers. Streszczenie. W artykule został przedstawiony zracjonalizowany algorytm wyznaczania iloczynu wektorowo-macierzowego, dla danych będących liczbami zespolonymi. Proponowany algorytm wyróżnia się w stosunku do metody...
In this paper a new approach to the optimized implementation of the Discrete Wavelet Transform (DWT) is presented, which is based on the original algorithm for performing basic DWT procedure with reduced number of multiplications. This approach allows reduction of requirements for hardware cost and computing time and creates usable conditions for e...
In this work a rationalized algorithm for calculating the product of sedenions is presented which reduces the number of underlying multiplications. Therefore, reducing the number of multiplications in VLSI processor design is usually a desirable task. The computation of a sedenion product using the naive method takes 256 multiplications and 240 add...
In this paper we introduce efficient algorithm for the multiplication of sedenions. The direct multiplication of two sedenions requires 256 real multiplications and 240 real additions. We show how to compute a sedenions product with 120 real multiplications and 344 real additions.
We consider algorithmic aspects of improving calculations of octonion product. Octonions together with quaternions represent a variety of hypercomplex numbers. An advantage of the suggested algorithm consists in decreased twice number of calculated real number products needed to compute the octonion product if compared to a straightforward naive wa...