A Lanczos algorithm for computing split quaternion partial singular value decomposition and its application

Abstract

Partial singular value decomposition (PSVD) is often used to deal with dimensionality reduction, compression and data approximation of large matrices to improve computational efficiency and simplify the analysis of complex data. In this paper, we will study the problem of PSVD of split quaternion matrix. By analyzing the structure of the real representation matrix, we find that it can preserve the partially unitary property. Based on the property of the real representation matrix, we propose split quaternion partial singular value decomposition (SQPSVD) and Lanczos algorithm of SQPSVD. In numerical examples, we verify the effectiveness of the Lanczos algorithm and propose the split quaternion color image model to apply the Lanczos algorithm to color image processing. The combination of the model and algorithm has a good performance in low rank approximation of color images, color face reconstruction and recognition. Moreover, based on the Lanczos algorithm, we propose the denoising algorithm for the space magnetic field measurement data and the experimental result shows the effectiveness of the denoising algorithm.