Classification by Principal Component Regression in the Real and Hypercomplex Domains

Linear regression is a simple and widely used machine learning algorithm. It is a statistical approach for modeling the relationship between a scalar variable and one or more variables. In this paper, a classification by principal component regression (CbPCR) strategy is proposed. This strategy depends on performing regression of each data class in terms of its principal components. This CbPCR formulation leads to a new formulation of the Linear Regression Classification (LRC) problem that preserves the key information of the data classes while providing more compact closed-form solutions. For the sake of image classification, this strategy is also extended to the 4D hypercomplex domains to take into account the color information of the image. Quaternion and reduced biquaternion CbPCR strategies are proposed by representing each channel of the color image as one of the imaginary parts of a quaternion or reduced biquaternion number. Experiments on two color face recognition benchmark databases show that the proposed methods achieve better accuracies by a margin of about 3% over the original LRC and like methods.