Randomized Quaternion Generalized Singular Value Decomposition Algorithms With Error Analysis

ABSTRACT In high‐dimensional data processing, quaternion generalized singular value decomposition (QGSVD) has become one of the most important solvers for advanced mathematical models. However, there are currently fewer efficient algorithms for calculating partial quaternion generalized singular values and vectors. When dealing with a quaternion matrix pair of substantial size, we present a stable QGSVD and two randomized algorithms cooperating with structure‐preserving and quaternion sampling strategies, and generate good low‐rank approximations to the original quaternion matrix pair. Explicitly, by leveraging random sampling based on the quaternion normal distribution for the original quaternion matrix pair, we propose both fixed‐rank and adaptive randomized algorithms for QGSVD. We also establish different kinds of error bounds for the randomized QGSVD approximation in the theoretical analysis and illustrate the effectiveness of the randomized QGSVD algorithms by numerical experiments on simulation data and practical color face recognition.