高维约束矩阵回归的优化理论与算法

High-dimensional constrained matrix regression refers to non-convex constraint statistical regression with the multivariate responses and multivariate predictors in the higher-dimensional setting. Its mathematical model is a matrix optimization, which is NP-hard and has a wide range of applications in areas such as machine learning and artificial intelligence, gene expression analysis, neural networks, medical imaging and diagnosis, risk management. This project focuses on optimization theory and algorithm of high-dimensional constrained matrix regression, which mainly contents the following parts: Firstly, we consider the statistical and optimization properties of high-dimensional matrix regression with equation, low-rank or sparse constraints and its various relaxations, and established high-dimensional model selection and parameter selection criteria and optimal conditions, which guarantee to solve relaxation problem easily and get the approximate effective errors; Secondly, we propose algorithms and give their convergent analyses which should be applied to solve large-scale matrix optimizations with fast and stable performance; Finally, we give numerical experiment and its application in medical imaging and diagnosis, and propose the practically useful programming. This projection will promote the cross integration between statistical and mathematical branches in operations research , and is valuable for rapid development of statistical optimization. Then it provides the theoretical and algorithmic foundations for solving problems in practice.

高维约束矩阵回归是指高维情况下带非凸约束的多响应多预测统计回归问题，其数学模型是一个NP-难的矩阵优化，它在机器学习与人工智能、医学影像疾病诊疗、基因表达分析、脑神经网络、风险管理等领域有广泛应用。本项目欲开展高维约束矩阵回归的优化理论与算法研究，主要内容包括: 探讨高维情况下带有等式和低秩稀疏等非凸约束的矩阵回归及各种松弛优化模型的统计和优化性质, 建立高维数据的模型选择标准和参数选取理论，以期给出各种矩阵优化模型的适用条件，保证松弛问题易求解、近似效果好且误差可控；进行算法设计与理论分析, 以期得到收敛速度快、稳定性能好、适合求解大规模矩阵优化问题的算法；进行数值实验及在医学影像疾病诊疗中的应用研究,以期得到具有实用价值的计算程序。本研究能促进统计学等多学科领域在最优化的融合与交叉，有利于统计优化的快速发展，为解决实际问题提供理论与算法支撑。

高维约束矩阵回归是指高维情况下带非凸约束的多响应多预测统计回归问题，其数学模型是一个NP-难的矩阵优化，它在机器学习与人工智能、医学影像疾病诊疗、基因表达分析、脑神经网络、风险管理等领域有广泛应用。本项目开展高维约束矩阵回归的优化理论与算法研究，主要内容包括: 探讨高维情况下带有等式和低秩稀疏等非凸约束的矩阵回归及各种松弛优化模型的统计和优化性质。针对高维统计模型，研究其对偶理论、一阶/二阶最优性条件、模型自由度、模型参数选择等，并分析模型的统计性质。在算法设计方面，针对凸/非凸稀疏优化模型，利用数据的结构特征，设计快速有效的优化算法，并探讨算法的收敛性分析。在实际应用方面，结合本项目所提出的模型和算法，将它们应用于视频监控的前景提取、工业过程的故障检测、磁共振成像的异常分析、人脸识别的特征选择等领域，进行数值实验及在医学影像疾病诊疗中的应用研究,得到了具有实用价值的计算程序。本研究促进统计学等多学科领域在最优化的融合与交叉，有利于统计优化的快速发展，为解决实际问题提供了理论与算法支撑。

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