Improved Analysis of Quaternion Randomized Block Krylov Iterations for Low‐Rank Matrix Approximation

ABSTRACT

This paper focuses on approximating a quaternion matrix and its singular values via the quaternion randomized block Krylov iteration (qRBKI) algorithm. Specifically, for qRBKI with block size , we establish novel gap‐based error bounds for the rank‐ matrix approximation in both Frobenius and spectral norms, explicitly demonstrating that qRBKI outperforms the quaternion randomized subspace iteration (qRSI) in approximation quality. For , leveraging simulated starting block vectors and quaternion polynomials, we show that the approximation error depends on the minimal relative singular value gap under the ‐order neighbouring structure. The near‐zero‐gap‐induced dependence can be eliminated by perturbing the input matrix with a low‐intensity random quaternion Gaussian or uniform matrix. Finally, numerical experiments on synthetic data and color image applications validate our theoretical findings and illustrate the effectiveness of qRBKI.