鲁棒线性鉴别分析理论及其一致稀疏性拓展

Linear discriminant analysis is one of the most simple and effective methods for feature extraction and classification. However, since the existing linear discriminant analysis theories and methods use the Euclidean distance as the basic measurement, they are sensitive to the noises and outliers. In order to solve this problem, this project proposes the novel ideas, i.e. robust locality preservation and robust weight projective reconstruction, to develop the robust linear discriminant analysis theory so as to obtain a more generalized robust discriminant analysis theory and algorithm framework based on the robust L2,1 norm. Since using the L2,1 norm as the robust measurement cannot obtain the sparse discriminant projection, we also utilize it as the regularizer and proposed robust consistent sparse discriminant analysis theory. We use the strategies of deduction and induction, contrastive study, searching for similarities and differences to perform theory analysis to reveal the robustness mechanism and the theoretical connections between the proposed algorithms and previous ones. This study will built a novel generalized discriminant analysis theorem and algorithm framework to significantly enhance the robustness of the discriminant analysis theories and methods, to further enrich the theoretical systems of feature extraction and to improve the computer’s analysis and understanding abilities on the variable level. The theories and algorithms proposed in this study can be widely used in multi-variable analysis, high-dimensional data processing, image feature extraction and recognition, and so on.

线性鉴别分析是当前特征提取与分类中最简单而有效的方法之一。然而，由于现有的线性鉴别分析理论与方法以欧式距离作为基本的度量，其对燥声与野点较敏感。为此，本项目以较鲁棒的L2,1范数作为基本的度量，提出了“鲁棒局部保持”、“鲁棒权重投影重构”等新思想来发展鲁棒线性鉴别分析理论，获得了更为广义的鲁棒鉴别分析理论与算法框架。而因单纯地把它作为度量并不能获得稀疏的鉴别向量，故本项目进而把L2,1范数作为正则项，提出鲁棒一致稀疏鉴别分析理论，利用“演义归纳，对比研究，寻找异同”的策略进行理论分析，揭示该系列算法的鲁棒性机制及其与现有方法的理论联系。本项目的研究将建立全新的广义鉴别分析理论与算法框架，显著增强鉴别分理论与方法的鲁棒性，进一步丰富特征提取的理论体系，提高计算机对模式变量层面上的语义分析与理解能力。本项目的研究成果在多变量数据分析，高维数据处理，图像特征提取与识别方面有着广泛的应用前景。

线性鉴别分析是当前特征提取与分类中最简单而有效的方法之一。然而，由于现有的线性鉴别分析理论与方法以欧式距离作为基本的度量，其对噪声与野点较敏感。为此，本项目以较鲁棒的L2,1范数作为基本的度量，提出了“鲁棒局部保持”、“鲁棒权重投影重构”等新思想来发展鲁棒线性鉴别分析理论，获得了更为广义的鲁棒鉴别分析理论与算法框架。而因单纯地把它作为度量并不能获得稀疏的鉴别向量，故本项目进而把L2,1范数作为正则项，提出鲁棒一致稀疏鉴别分析理论。具体地，在项目研究期间，我们提出了鲁棒鉴别回归、鲁棒局部鉴别分析和旋转不变维数约简等一系列方法。项目期间发表了总计17篇学术论文，其中包含11篇transactions系列文章。本项目的研究建立全新的广义鉴别分析理论与算法框架，显著增强鉴别分理论与方法的鲁棒性，进一步丰富特征提取的理论体系，提高计算机对模式变量层面上的语义分析与理解能力。本项目的研究成果在多变量数据分析，高维数据处理，图像特征提取与识别方面有着广泛的应用前景。