Perceptron Research Based On Clifford Algebra

Clifford algebra which combine inner and outer product has many applications ingeometry and physics. Clifford algebra for machine learning as a research focus has abroad prospects. In order to solve the problem that the classifier can only be divided intotwo categories, Clifford algebra is applied to the classifier. In this paper, the Cliffordalgebra as a mathematical tool is used in the model of the perceptron.Firstly, we researched the Clifford perceptron on the basis of that Clifford algebra isapplied to the model of the perceptron. We get the discriminant function of the Cliffordperceptron by training the Clifford data. Then, we conduct classification of themulti-class according to the obtained Clifford value of the discriminant function. In orderto evaluate the performance of the Clifford perceptron, we used three groups of UCI dataset and four groups of BCI data set to carry on the simulation experiment, and comparedthe classification method of the Clifford perceptron with the others. The results of theexperiment show that this algorithm can achieve satisfactory results.Secondly, we proposed the Clifford kernel perceptron on the basis of that Cliffordalgebra is applied to the model of the kernel perceptron. Using the Clifford kernelfunction map the Clifford data to the Clifford kernel space, and then apply the model ofthe Clifford perceptron in the Clifford kernel space. In order to evaluate the performanceof the Clifford kernel perceptron, we used two groups of UCI data set and one group ofBCI data set to carry on the simulation experiment, and compared the classificationmethod of the Clifford kernel perceptron with the others. The results of the experimentshow that this algorithm can achieve satisfactory results.Thirdly, because different feature sequencing will lead to different Clifford data, weconduct feature sequencing by using the algorithm of the differential evolution. In orderto evaluate the impact of the feature sequencing for the classification results, we based onthe Clifford perceptron’s classification results to obtain the optimal feature sequencing.we used one group of BCI data set to carry on the simulation experiment, and the results of the experiment show that the algorithm of the differential evolution for featuresequencing is feasible.Finally, the Clifford perceptron is extended to multilayer, that is Clifford multilayerperceptrons(MLPs). We introduced the frame of Clifford MLPs, the universalapproximation theory, the activation functions and the Clifford back-propagationalgorithm in the space of the Clifford algebra.