Efficiency parallel computing using quaternious

Abstract

The article presents a systematic study and comparative analysis of the efficiency of parallel computing when operating with non-commutative quaternion data structures and solving quaternion linear systems Ax=b on heterogeneous hardware platforms (CPU, GPU, TPU) using modern software stacks (NumPy, TensorFlow Graphics, tfquaternion, CuPy). The theoretical aspects of using Hamilton algebra for processing multidimensional and multichannel signals (color spaces, 3D orientation) are considered, which provides a reduction in the number of independent training parameters of neural network models by 75%. The computational overhead of graphs is analyzed, due to the lack of native hardware support for non-commutative multiplication and the need to simulate the Hamiltonian product through 16 real operations and a real block representation (Real Matrix Representation) with dimension 4M x 4N. Using the developed benchmarking software tool in the Google Colab cloud infrastructure, the latency of the Hamiltonian product execution for samples of size N in [10^2...10^6] and the speed of LU-decomposition of linear systems are estimated. The effect of using XLA-compilation (kernel fusion) and FP16-computing when optimizing QCNN graphs is studied using the example of image classification from the CIFAR-10 and HAM10000 datasets. Practical recommendations are formed for choosing a software-hardware stack depending on the data dimension and computational latency requirements.

Keywords: quaternions, Hamiltonian product, parallel computing, QCNN, TensorFlow Graphics, CuPy, XLA compilation, GPU, TPU, real matrix representation.