One-Pass Randomized Algorithm with Practical Rangefinder for Low-Rank Approximation to Quaternion Matrices

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.