标题： 高效随机算法用于四元数矩阵的低秩逼近 显示英文标题

标题： Pass-efficient Randomized Algorithms for Low-rank Approximation of Quaternion Matrices

摘要： 随机化算法用于四元数矩阵的低秩逼近近年来受到越来越多的关注。然而，现有方法忽略了遍历效率，即限制对输入矩阵的遍历次数的能力——这在由通信成本主导的现代计算环境中至关重要。我们通过提出一系列遍历高效的随机算法来弥补这一差距，使用户能够直接将遍历预算与逼近精度进行权衡。我们的贡献包括：(i) 一种针对四元数矩阵低秩逼近的任意遍历随机算法族，在用户指定的矩阵视图数量下运行，以及 (ii) 一种遍历高效的块Krylov子空间方法，可加速谱衰减缓慢的矩阵的收敛。此外，我们建立了谱范数误差界，表明期望的逼近误差随着遍历次数的增加呈指数衰减。最后，我们通过广泛的数值实验验证了我们的框架，并展示了其在多个应用中的实际相关性，包括四元数数据压缩、矩阵补全、图像超分辨率和深度学习。 显示英文摘要

摘要： Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input matrix-which is critical in modern computing environments dominated by communication costs. We address this gap by proposing a suite of pass-efficient randomized algorithms that let users directly trade pass budget for approximation accuracy. Our contributions include: (i) a family of arbitrary-pass randomized algorithms for low-rank approximation of quaternion matrices that operate under a user-specified number of matrix views, and (ii) a pass-efficient extension of block Krylov subspace methods that accelerates convergence for matrices with slowly decaying spectra. Furthermore, we establish spectral norm error bounds showing that the expected approximation error decays exponentially with the number of passes. Finally, we validate our framework through extensive numerical experiments and demonstrate its practical relevance across multiple applications, including quaternionic data compression, matrix completion, image super-resolution, and deep learning.