学习理论中的核典型相关分析及相关算法的研究和应用

Canonical correlation analysis (CCA)is an important data feature extraction method, which focuses on mutual information that shared by two sets of multidimensional variables. It has been widely used in multimedia information retrieval, human and machine interface, gene sequence analysis, text recognition and some other fields. But there is not much known about theoretical background of kernel CCA (kernel version of CCA). In this project, we shall conduct theoretical analysis about kernel CCA based on least squares regression and coefficient-based regularization in statistical learning theory. A reduced-order model based online kernel CCA algorithm will be addressed. The convergence rates and sparsity of the algorithm for multiple conditional CCA with indefinite kernels will be elucidated, too. We will also study the consistency of conditional kernel CCA both in one-dimension and multi-dimension (dimension of feature spaces). Meantime, we will apply distance correlation and elastic net kernel CCA to study functional connectivity in ROI (region of interest) of brain. The research results will give insight to a better understanding about the relationship between variation of functional connectivity in specific brain region and corresponding abnormal human behavior.

典型相关分析是一种研究两个多维变量集之间共同特征的特征提取算法。它在多媒体信息检索、人机交互、基因序列分析、文本识别等领域有着广泛应用。然而，核典型相关分析(引入核方法的典型相关分析)数学理论的研究仍处于初级阶段。本项目将运用统计学习理论中已有的关于最小二乘回归，系数正则化的理论来研究核典型相关分析的相关理论。本项目研究降阶模型下，在线算法和核典型相关分析的融合；研究不定核多重核典型相关分析的收敛速率及算法稀疏性；给出条件核典型相关分析在一维和多维情形一致性的刻画指标。本项目将同时应用距离相关的理论以及弹性网格核典型相关分析的理论研究大脑感兴趣区域的功能连接。研究所得成果将揭示脑区功能连接异常对人类某些变异行为变化的影响。

典型相关分析是一种研究两个多维变量集之间共同特征的特征提取算法。它在多媒体信息检索、人机交互、基因序列分析、文本识别等领域有着广泛应用。然而，核典型相关分析(引入核方法的典型相关分析)数学理论的研究仍处于初级阶段,相应算法的研究仍在不断完善之中。本项目运用统计学习理论中已有的关于最小二乘回归，系数正则化，稀疏性，鲁棒性等理论研究了核典型相关分析的相关理论和一系列算法。主要包括：运用梯度下降算法研究核典型相关分析，提出了鲁棒，快速核典型相关分析算法，研究了ERM情形带噪声情形的核典型相关分析算法。提出了条件核典型相关分析算法。我们同时研究了稀疏分位数回归的一致性分析。所得研究成果极大促进了数据相关性分析理论，算法的发展。对信息检索，文本识别等领域的发展起到了重大的促进作用。

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