结构矩阵的低秩逼近及其应用

项目名称： 结构矩阵的低秩逼近及其应用

项目编号： No.11271259

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 刘永辉

作者单位： 上海对外经贸大学

项目金额： 60万元

中文摘要： 利率衍生品定价和资产收益的因子模型给数值代数提出了许多亟待解决的问题，这些问题都与具有某种结构的秩约束矩阵逼近问题有关。本项目研究结构矩阵的低秩逼近及其应用。针对相关矩阵的低秩逼近及其推广，我们将根据结构相关矩阵的特点，提出新的处理"秩约束"的手段和方法，使用矩阵分解、矩阵分块等数值代数预处理技术和序列线性方程组等非线性优化方法研究该问题的收敛性，寻求速度快、且计算稳定的新算法；对于降秩的多元回归模型和增长曲线模型，我们将使用联合矩阵分解、核范数、优化函数等方法处理秩约束，继而研究模型的快速计算和统计检验问题。本项目不仅对矩阵低秩逼近等计算数学的理论和方法有所贡献，而且对金融市场也具有重要的实际应用价值。

中文关键词： 结构低秩逼近；矩阵分解；约束优化方法；金融统计诊断；信用衍生品定价

英文摘要： Many numerical linear algebra problems come from interest rate derivatives pricing models and factor models of asset returns, and these problems involve matrix approximation with low rank and special structures. In this project, we will investigate the structured low-rank approximation with applications. At first, we will propose some new methods to deal with "low-rank of correlation matrix" based on the features of correlation matrix, and we will use the tricks of matrix decomposition, block matrix and sequential linear equations to seek some new algorithms and study its convergence. Secondly, we will apply the tricks of the simultaneous matrix decompositions, nuclear norm and majorization function to deal with the low-rank of reduced rank regression model and reduced rank growth curve model, and propose some new algorithms and study its statistical test. Through the project, not only can we make some contributions to the low-rank matrix approximation in numerical mathematics, but also provide theoretical guide for practical aspects in finance.

英文关键词： Structured low-rank approximation；matrix decomposition；constrained optimization methods；financial statistics diagnostics；pricing credit default swap