As its real/complex counterparts, randomized algorithms for low-rank approximation to quaternion matrices received attention recently. For large-scale problems, however, existing quaternion orthogonalization methods are not efficient, leading to slow rangefinders. By relaxing orthonormality while maintaining favaroable condition numbers, this work proposes two practical quaternion rangefinders that take advantage of mature scientific computing libraries to accelerate heavy computations. They are then incorporated into the quaternion version of a well-known one-pass algorithm. Theoretically, we establish the probabilistic error bound, and demonstrate that the error is proportional to the condition number of the rangefinder. Besides Gaussian, we also allow quaternion sub-Gaussian test matrices. Key to the latter is the derivation of a deviation bound for extreme singular values of a quaternion sub-Gaussian matrix. Numerical experiments indicate that the one-pass algorithm with the proposed rangefinders work efficiently while only sacrificing little accuracy. In addition, we tested the algorithm in an on-the-fly 3D Navier-Stokes equation data compression to demonstrate its efficiency in large-scale applications.

翻译：与实/复矩阵类似，针对四元数矩阵低秩近似的随机算法近年来受到关注。然而，在处理大规模问题时，现有四元数正交化方法效率低下，导致寻径器计算缓慢。本文通过放宽正交性条件并保持良好条件数，提出两种实用四元数寻径器，借助成熟科学计算库加速重计算。将其融入经典单遍算法的四元数版本后，理论上推导出概率误差界，并证明误差与寻径器条件数成正比。除高斯矩阵外，我们还允许使用四元数次高斯测试矩阵——其关键技术在于推导四元数次高斯矩阵极端奇异值的偏差界。数值实验表明，采用所提寻径器的单遍算法在牺牲极少精度的前提下高效运行。此外，我们将该算法应用于实时三维纳维-斯托克斯方程数据压缩，验证其在大规模应用中的高效性。