将多种随机草图整合用于奇异值分解
Integrating multiple random sketches for singular value decomposition
AI总结：
本文提出基于多种随机草图的蒙特卡洛类型整合SVD算法，通过优化问题与矩阵Stiefel流形约束，提高SVD精度与稳定性，理论分析证明其一致性，数值结果表明其有效性。
摘要：
大规模矩阵的奇异值分解（SVD）是数据科学和科学计算中的关键工具。随着矩阵规模的快速增长，开发高效的大规模SVD算法需求增加。基于一次随机草图的随机化SVD已被研究，并展示了计算低秩SVD的潜力。本文提出一种基于多种随机草图的蒙特卡洛类型整合SVD算法。该算法通过整合多个随机草图子空间的结果，从而实现更高的精度和更低的随机波动。整合的核心是一个具有矩阵Stiefel流形约束的优化问题。该优化问题通过Kolmogorov-Nagumo型平均解决。理论分析表明，通过总体平均可以诱导奇异向量，并确保计算和真实子空间及奇异向量之间的一致性。统计分析进一步证明了大数定律，并通过中心极限定理给出了收敛速率。初步的数值结果表明，所提出的整合SVD算法具有前景。
英文摘要：
The singular value decomposition (SVD) of large-scale matrices is a key tool in data analytics and scientific computing. The rapid growth in the size of matrices further increases the need for developing efficient large-scale SVD algorithms. Randomized SVD based on one-time sketching has been studied, and its potential has been demonstrated for computing a low-rank SVD. Instead of exploring different single random sketching techniques, we propose a Monte Carlo type integrated SVD algorithm based on multiple random sketches. The proposed integration algorithm takes multiple random sketches and then integrates the results obtained from the multiple sketched subspaces. So that the integrated SVD can achieve higher accuracy and lower stochastic variations. The main component of the integration is an optimization problem with a matrix Stiefel manifold constraint. The optimization problem is solved using Kolmogorov-Nagumo-type averages. Our theoretical analyses show that the singular vectors can be induced by population averaging and ensure the consistencies between the computed and true subspaces and singular vectors. Statistical analysis further proves a strong Law of Large Numbers and gives a rate of convergence by the Central Limit Theorem. Preliminary numerical results suggest that the proposed integrated SVD algorithm is promising.
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