本项目拟研究矩阵分解的随机算法、随机扰动分析及其应用。首先，针对大型低秩矩阵，利用与各种结构随机矩阵相结合的随机投影方法、与重要抽样分布相结合的随机抽样方法、小型矩阵分解的传统算法等，设计部分经典矩阵分解、双曲矩阵分解、矩阵对的矩阵分解的随机算法，并利用传统的误差分析、矩阵分解的(随机)扰动分析与概率统计理论等探讨算法的误差分析；其次，利用矩阵方程方法、矩阵向量方程方法、Lyapunov 受控函数技术、不定点定理、算子分拆技术等，并结合概率统计的相关结论(如Markov 不等式、Chebyshev不等式、Slutsky 定理等)与随机矩阵的相关性质等，研究各种矩阵分解在随机扰动下基于平均意义与概率意义的各类型的条件数与扰动界，并利用概率谱范数估计、小样本统计条件估计等探讨条件数与扰动界的统计估计与算法；最后，结合上述结果探讨(约束)不定最小二乘问题的随机算法与随机扰动分析。

This project will investigate the stochastic algorithms and stochastic perturbation analysis of matrix factorizations and their applications. Firstly, for large matrix with low rank, we will design the stochastic algorithms of some classic matrix factorizations, hyperbolic matrix factorizations, and matrix factorizations of matrix pencil using the random projection methods with some structured random matrices, the random sampling methods with the importance sampling distribution, and the regular algorithms for matrix factorizations of small matrix, and discuss their error analysis using the theory of regular error analysis, the theory of (stochastic) perturbation analysis for matrix factorizations, and the theory of probability-statistics. Then, with the matrix equation approach, the matrix-vector equation approach, the technique of Lyapunov majorant function, the fixed point theorems, and the technique of splitting operators, and combining some results of probability-statistics (such as the Markov inequality, the Chebyshev inequality, and the Slutsky theorem) and the properties of random matrix, we will study all types of condition numbers and perturbation bounds in the average and probabilistic senses of matrix factorizations under the stochastic perturbation, and consider the statistics estimates and algorithms of the obtained condition numbers and perturbation bounds using the probabilistic spectral norm estimator and the small-sample statistical condition estimation. Finally, the stochastic algorithms and stochastic perturbation analysis of the (constrained) indefinite least squares problem will be discussed based on the above results.