稀疏优化问题的理论与方法及其应用

This proposal presents the theory analysis and algorithm design of sparse optimization. Form the views of vector and matrix, a variety of sparse models will be constructed and applied to pattern recognition for high dimensional data. It mainly contains: 1) According to the biological features of gene expression data, higher dimension but lower sampling, a special sparse optimization is introduced to represent the inter-relation between genes. Under two classes independence rule, the most discriminative features are determined in the optimal and statistical senses. 2) A generalized l2,p(0<p<=1)-norm minimizations will be considered and a unified algorithm is proposed, also the involved convergence. The results provide algorithmic support to adaptively choose better sparse model for different sparse data structures. 3) Based on the theoretical results about l1-mimization problems, also considering the matrix norm nature, this proposal will discuss the joint sparsity, computational complexity and robustness of non-convex and non-Lipschitz continuous l2,p(1<p<=1) based minimization problem.4) To overcome the inefficiency of determining goals one by one, a joint sparse model is constructed with different distrbution pattern. Based on the conclusions in 2) and 3), a unified algorithm and its convergence analysis will be presented. The new algorithm will be also applied to robust face recognition.

本项目将提供稀疏优化问题的理论分析和算法设计，从向量、矩阵两种角度建立模型，并将之应用于高维数据的模式识别。主要内容有：1)针对生物基因表达数据高维数、低采样的特点，引入稀疏优化模型表示基因间的交互关系，结合统计两分类t-test和独立法则，确定出优化和统计意义下最具识别能力的基因。2)为适应不同的稀疏结构，建立广义的混合l2,p(0<p<=1)-模 极小化模型，提出统一求解的算法，并证明收敛性，这一结果为自适应选取稀疏优化模型提供了算法基础。3)从向量l1-模极小化的一系列结论出发，融合矩阵范数的本质，建立非凸、非Lipschitz连续的混合l2,p(0<p<=1) -模极小化问题解的联合稀疏性、计算复杂度和鲁棒性等方面的理论框架。4)为了克服逐个目标识别的单一性，建立具有不同分布方式的联合稀疏表示模型，基于2)、3)的结论设计统一求解算法并给出收敛性分析，进而应用于鲁棒人脸识别。

最优化理论与方法在信息科学、统计学、地球物理学、经济学等自然科学和社会科学领域有着广泛的应用，由于各领域的学科背景与数据特点不同，产生的优化问题也具有各式各样的特殊结构。本项目考虑的特殊优化问题来自于机器学习领域的集体图像识别。由于大多数图像数据都具有一定的稀疏特点，比如格式上的稀疏性或者知识表达的稀疏性，充分利用数据稀疏性这一先验信息的稀疏学习，将大大提高学习的功能和效率。本项目构建的优化模型采用矩阵而不是传统的向量为表示变量，不仅使大规模图像的集体处理更加快捷有效，而且体现了各图像集之间的相互关系。而矩阵变量的稀疏性度量以及稀疏矩阵优化问题的有效求解算法则是本项目的主要研究目标。