矩阵的结构主成份分析及其应用

本课题旨在探索矩阵数据的维度约简与特征提取的一种新技术，称为结构主成份分析（S-PCA）。S-PCA的基本思想是：首先，对矩阵空间进行互补结构分解得到低维结构子空间，每个结构子空间有各自的几何结构；然后，矩阵数据被分解到结构子空间，并进行保持结构的PCA。S-PCA的优点在于：（1）结构子空间的低维度使得S-PCA比标准PCA有更高的计算效率；（2）保持结构的PCA使得S-PCA比标准PCA有更好鲁棒性；（3）与K-PCA正好相反，S-PCA是通过结构分解将高维数据变换到低维结构子空间，并且在重构过程中不涉及任何非线性优问题；（4）就维度约简而言，在相同的信息损失率下S-PCA比GLRA具有更低的约简维度。本课题的研究内容包括：S-PCA算法、对称型的S-PCA、结构选择和S-PCA在计算机视觉中的应用。

This proposal is to study a novel method for dimensionality reduction and feature extraction of data in matrix form,called Structural Principal Component Analysis (S-PCA).The basic idea of S-PCA is: First, matrix space is decomposed into some low-dimensional subspaces, each of which has some different geometrical structure, by a complementary structures decomposition,then matrix data are mapped into the structural subspaces,and a PCA with structure preserving property is carried out. S-PCA has the following advantages:(1) The low dimensionality of the structure subspaces makes it more computational efficient than the standard PCA. (2) The PCA with structure preserving makes it more robust than the standard PCA.(3) In contrast to the K-PCA, S-PCA transforms the high-dimensional data into the low-dimensional structural subspaces by the structures decomposition, and does not involve any nonlinear optimization in the reconstruction process.(4) S-PCA has lower dimensionality than the GLRA does under the same information loss rate. The key issues of this proposal include: S-PCA algorithm, Symmetrical S-PCA, Structure selection, and applications of S-PCA in computer vision.