A class of multidimensional NIPALS algorithms for quaternion and tensor partial least squares regression

Abstract

Abstract Quaternion and tensor-based signal processing benefits from exploiting higher dimensional structure in data to outperform the corresponding approaches using multivariate real algebras. Along with the extended range of processing options, these methods produce opportunities for a physically-meaningful interpretation. In this paper, we propose a class of novel partial least squares (PLS) algorithms for tensor- and quaternion-valued data, the widely linear quaternion PLS (WL-QPLS), the higher order nonlinear iterative PLS (HONIPALS) and the generalised higher order PLS (GHOPLS). This enables a regularised regression solution along with a latent variable decomposition of the original data based on the mutual information in the input and output block. The performance of the proposed algorithms is verified through analysis, together with a detailed comparison between quaternions and tensors and their application for image classification.