QR and LU factorizations are two basic mathematical methods for decomposition and dimensionality reduction of large-scale matrices. However, they are too complicated to be executed for a limited client because of big data. Outsourcing computation allows a client to delegate the tasks to a cloud server with powerful resources and therefore greatly reduces the client’s computation cost. However, the previous methods of QR and LU outsourcing factorizations need multiple interactions between the client and cloud server or have low accuracy and efficiency in large-scale matrix applications. In this paper, we propose a noninteractive and efficient outsourcing algorithm of large-scale QR and LU factorizations. The proposed scheme is based on the specific perturbation method including a series of consecutive and sparse matrices, which can be used to protect the original matrix and obtain the results of factorizations. The generation and inversion of sparse matrix has small workloads on the client’s side, and the communication cost is also small since the client does not need to interact with the cloud server in the outsourcing algorithms. Moreover, the client can verify the outsourcing result with a probability of approximated to 1. The experimental results manifest that as for the client, the proposed algorithms reduce the computational overhead of direct computation successfully, and it is most efficient compare with the previous ones.

中文翻译：

---

大规模 QR 和 LU 分解的高效非交互式外包

QR 和 LU 分解是用于大规模矩阵分解和降维的两种基本数学方法。然而，由于大数据，它们太复杂而无法为有限的客户执行。外包计算允许客户端将任务委托给具有强大资源的云服务器，从而大大降低了客户端的计算成本。然而，之前的QR和LU外包分解方法需要客户端和云服务器之间的多次交互，或者在大规模矩阵应用中准确性和效率较低。在本文中，我们提出了一种非交互式且高效的大规模 QR 和 LU 分解的外包算法。所提出的方案基于特定的扰动方法，包括一系列连续和稀疏的矩阵，可用于保护原始矩阵并获得因式分解的结果。稀疏矩阵的生成和求逆在客户端的工作量小，通信成本也小，因为在外包算法中客户端不需要与云服务器交互。此外，客户端可以以近似为1的概率验证外包结果。实验结果表明，对于客户端，所提出的算法成功地降低了直接计算的计算开销，并且与之前的算法相比效率最高。并且由于客户端不需要在外包算法中与云服务器交互，因此通信成本也很小。此外，客户端可以以近似为1的概率验证外包结果。实验结果表明，对于客户端，所提出的算法成功地降低了直接计算的计算开销，并且与之前的算法相比效率最高。并且由于客户端不需要在外包算法中与云服务器交互，因此通信成本也很小。此外，客户端可以以近似为1的概率验证外包结果。实验结果表明，对于客户端，所提出的算法成功地降低了直接计算的计算开销，并且与之前的算法相比效率最高。