Abstract:

Recently, indefinite kernel support vector machine (IKSVM) has attracted great attention in the machine learning community as more and more indefinite metric kernel matrices have occurred. However, the existing IKSVM algorithms are usually unable to solve the problems of information redundancy and sample sparsity caused by high-dimensional data. In view of the research status, the existing mainstream IKSVM algorithms are studied.Based on the stabilization problem of IKSVM in the reproducing kernel Kre?n spaces (RKKS), this paper proves theoretically that the essence of the IKSVM is the sequential application of indefinite kernel principal component analysis (IKPCA) and support vector machine (SVM) in reduced spaces, and proposes a novel learning framework for IKSVM: two-phase indefinite kernel support vector machine (TP-IKSVM). The TP-IKSVM solves the IKSVM problem by applying IKPCA and SVM successively, combining the advantages of IKPCA in alleviating redundancy and sample sparsity caused by high-dimensional datasets and the good generalization performance of SVM in reduced spaces. Experimental results on real world datasets show the classification accuracy of TP-IKSVM is better than that of the existing mainstream IKSVM algorithms.

Key words: indefinite kernel, reproducing kernel Kre?n spaces (RKKS), indefinite kernel principal component analysis (IKPCA), indefinite kernel support vector machine (IKSVM)

摘要：

近年来，在机器学习的各个领域出现了越来越多不定的度量核矩阵，使得不定核支持向量机（IKSVM）得到了广泛关注。但是，现有IKSVM算法通常不能较好地解决高维数据所带来的信息冗余和样本稀疏等问题。针对此研究现状，对现有主流的IKSVM算法进行了研究，并基于再生核Kre?n空间（RKKS）中对IKSVM问题的稳定化定义，从理论上证明了IKSVM问题的本质为不定核主成分分析（IKPCA）降维后空间中的支持向量机（SVM）问题，进一步地提出求解IKSVM问题的新型学习框架TP-IKSVM。TP-IKSVM通过将IKSVM问题的求解拆分为IKPCA和SVM两个阶段，充分地发挥了IKPCA在处理高维数据的信息冗余和样本稀疏等方面的优势，同时结合SVM以有效分类。在真实数据集上的实验结果表明，TP-IKSVM的分类精度优于现有主流的IKSVM算法。

关键词: 不定核, 再生核Kre?n空间（RKKS）, 不定核主成分分析（IKPCA）, 不定核支持向量机（IKSVM）