多层次结构化矩阵低秩逼近与稀疏表示方法建模与算法研究

The theories of low-rank decomposition, structured decomposition and sparse representation of a matrix have been widely used in a great deal of scientific and engineering fields. In recent years, the relevant theories have been deeply studied and made great progress. The project mainly focuses on: (1) the existence of solutions and algorithms for the low-rank approximation model with multi-level and multi-structural constraints and the low-rank approximation model with banded structure constraints, the algorithms for the multi-objective structured low-rank approximation model under different metrics and stability of the solutions under noise perturbations, (2) giving the best approximation solution of rank k to the structured low rank sparse representation problem by using the matrix polynomial of rank 1, reconstructing the low-rank part and sparse part of a observation matrix with defect data, studying the existence of solutions and algorithms for the structured low- rank approximation model with sparse noise, (3) construction of preconditioned algorithms for solving the linear complementary problem with its coefficient matrix belonging to a special class of matrix and studying the stability of the preconditioned algorithms with sparse noise, construction of the new complementary function and structure of complementary standards and studying the structure solutions and sparse solutions under the above standards. The research results of the project will provide efficient algorithms and theoretical basis for solving the problem of multi-level structured low-rank approximation of a matrix, the model of structured low-rank sparse representation and structured sparse solution of linear and nonlinear complementary problem.

矩阵的低秩分解、结构化分解以及稀疏表示理论在众多科学工程领域中都有广泛的应用。近年来，相关理论得到深入研究并取得重大进展。本项目主要研究：（1）多层次、多结构低秩逼近模型和带带状约束条件的低秩稀疏表示模型解的存在性和算法；不同度量标准下多目标结构化低秩逼近模型求解算法及其在噪声扰动下解的稳定性；（2）用秩1矩阵多项式给出结构化低秩稀疏表示问题秩为k的最佳逼近解；重构具有缺损数据的观测矩阵的低秩和稀疏部分；研究带有稀疏噪声结构化低秩逼近模型解的存在性和求解算法。（3）构建求解系数矩阵为特殊类矩阵线性互补问题的预处理算法，并研究稀疏噪声干扰下预处理算法的稳定性；构建互补问题新的互补度函数和结构互补标准，并研究此标准下互补问题的结构解和稀疏解。项目的研究成果将为求解多层次结构化矩阵的稀疏低秩逼近问题、结构化低秩稀疏表示模型和线性与非线性互补问题的结构化低秩、稀疏解提供有效算法和理论基础。