高维混合非线性结构数据分析及其应用

Analyzing the high-dimensional data with complex structure has become a hot topic in the field of information sciences in recent years. Current methods such as manifold learning, sparse representation, subspace clustering, etc usually make strong assumptions about the structure of the data, for example, one or multiple subspaces, one manifold. However, the data sampled from the real-world applications often have the following features: nonlinear, piece-wise smooth, different dimensions and multiple manifolds. This project will focus on the analysis of data with the structure of the mixing of multiple manifolds，use the effective tools such as Veronese mapping, Grassmann manifold, tensor bundle, Kernel methods, sparse and low-rank representation to research the structure-constrained based low-rank multi-manifolds representation model, metric on the feature space, the sparsity of the data in the feature space, the application of the nonlinear manifold clustering in computer vision and computer graphics. This project plans to provide theory methods to analyze the data with the structure of the mixing of multiple manifolds and apply them into the problems in visual analysis and digital geometry, in addition, provide the theoretical and algorithmic support for big data analysis. In summary, focusing on the basic research of the core issue in data analysis and including basic theory in different fields and new information processing technology, this project not only has some difficulty but also has meaningful significance in both theory and application.

近年来，高维复杂结构数据分析已成为信息科学领域研究的重点。现有的数据分析方法，如流形学习、稀疏表示、子空间聚类等往往需要对数据的结构给出较强约束，如单一或多个线性子空间或单一流形等。然而真实场景采样的数据往往具有非线性，分片光滑、不同维数和多流形等特点。本项目拟围绕混合多流形结构数据的分析问题展开研究，利用Veronese映射，Grassmann流形，张量丛，核方法，稀疏与低秩等工具，研究基于结构约束的多流形低秩表示、特征空间的度量、数据在变换空间中的稀疏性、非线性流形聚类在视觉分析和几何模型分析和处理中的应用。本项目拟给出能够分析具有多流形结构数据的理论方法，并应用到视觉分析和数字几何中的相关问题，对更一般的大规模高维数据的分析提供理论和算法支撑。本项目涉及多个领域的基础理论以及最新的信息处理技术，面向数据分析的核心问题进行应用基础研究，具有一定的难度，既有理论意义，也极具应用价值。

近年来，高维复杂结构数据分析已成为信息科学领域研究的重点。现有的数据分析方法，如流行学习、稀疏表示、子空间聚类等往往需要对数据的结构给出较强的约束，如单一或多个线性子空间或单一流形等。然而真实场景采样的数据往往具有非线性，分片光滑、不同维数和多流形等特点。本项目拟围绕混合多流行结构数据的分析问题开展研究，利用Vernese映射，Grassmann流形，张量丛，核方法，稀疏与低秩等工具，研究基于结构约束的多流形低秩表示、特征空间的度量、数据再变换空间中的稀疏性、非线性流行聚类在视觉分析和几何模型分析和数字几何中的相关问题，对更一般的大规模高维数据的分析提供理论和算法支撑。本项目涉及多个领域的基础只是以及最新的信息处理技术，面向数据分析的核心问题进行应用基础研究，具有一定的难度，既有理论意义，也极具有应用价值。