Genetic Algorithms for the Optimization of Quaternion Convolutional Neural Networks: A Review

Abstract

Quaternion convolutional neural networks (QCNNs) extend real-valued CNNs into the four-dimensional
quaternion algebra, treating the color channels of an image as a single hypercomplex entity. Through the
Hamilton product, QCNN filters couple all input channels simultaneously, capturing inter-channel
relationships that real-valued networks process independently, and typically achieving comparable
accuracy with roughly four times fewer trainable parameters. Training these networks by gradient
descent, however, inherits a fundamental obstacle of hypercomplex analysis: the loss is a real-valued
function of quaternion variables and is therefore nowhere analytic in the sense of the Cauchy–Riemann–
Fueter equations, so quaternion derivatives must be replaced by pseudo-gradients computed component-
wise, for which the product and chain rules of ordinary calculus fail. Gradient-free optimization, and in
particular the genetic algorithm (GA), offers a principled way to sidestep this obstacle entirely: the
network is trained by selection, crossover, and mutation acting on populations of weight configurations,
requiring only forward evaluations of the loss. This article reviews the intersection of these two lines of
research. We summarize quaternion algebra and the building blocks of modern QCNNs; examine the
analytic difficulties of quaternion backpropagation; survey the use of genetic algorithms in neural
network optimization, from early weight evolution through modern deep neuroevolution and evolutionary
architecture search; and review the emerging literature applying metaheuristic and evolutionary
optimization to quaternion-valued networks. We then organize the design space of GAs for QCNN
optimization—chromosome encodings for quaternion weights, quaternion-aware crossover and mutation
operators, fitness assignment, and hybrid GA–gradient schemes—and identify open problems, including
scalability of population-based search to convolutional-scale parameter counts and the need for
quaternion-native evolutionary operators.