ISSN 1000-1239 CN 11-1777/TP

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基于复随机样本的结构风险最小化原则

哈明虎1 田景峰2 张植明1   

  1. 1(河北大学数学与计算机学院 河北保定 071002) 2(华北电力大学科技学院 河北保定 071051) (minghuha@yahoo.com.cn)
  • 出版日期: 2009-11-15

Structural Risk Minimization Principle Based on Complex Random Samples

Ha Minghu1, Tian Jingfeng2, and Zhang Zhiming1   

  1. 1(College of Mathematics and Computer Sciences, Hebei University, Baoding, Hebei 071002) 2(College of Science and Technology, North China Electric Power University, Baoding, Hebei 071051)
  • Online: 2009-11-15

摘要: 统计学习理论目前是处理小样本学习问题的最佳理论.然而,该理论主要是针对实随机样本的,它难以讨论和处理现实世界中客观存在的涉及复随机样本的小样本统计学习问题.结构风险最小化原则是统计学习理论的核心内容之一,是构建支持向量机的重要基础.基于此,研究了基于复随机样本的统计学习理论的结构风险最小化原则.首先,给出了标志复可测函数集容量的退火熵、生长函数和VC维的定义,并证明了它们的一些性质;其次,构建了基于复随机样本的学习过程一致收敛速度的界;最后,给出了基于复随机样本的结构风险最小化原则,证明了该原则是一致的,同时推导出了收敛速度的界.

关键词: 统计学习理论, 结构风险最小化原则, 支持向量机, 复随机样本, VC维

Abstract: Statistical learning theory is commonly regarded as a sound framework that handles a variety of learning problems in presence of small size data samples. However, statistical learning theory is based on real random samples (real number valued random variables) and as such is not ready to deal with the statistical learning problems involving complex random samples (complex number valued random variables), which can be encountered in real world scenarios. Structural risk minimization principle is one of the kernel contents of statistical learning theory. The principle is the theoretical fundamental of establishing the support vector machine. Based on the above, the structural risk minimization principle of statistical learning theory based on complex random samples is explored. Firstly, the definitions of annealed entropy, growth function and VC dimension of a set of complex measurable functions are proposed and some important properties are proved. Secondly, the bounds on the rate of uniform convergence of learning process based on complex random samples are constructed. Finally, the structural risk minimization principle of complex random samples is proposed. The consistency of this principle is proven, and asymptotic bounds on the rate of convergence are derived. The investigations will help lay essential theoretical foundations for establishing support vector machine based on complex random samples.

Key words: statistical learning theory, structural risk minimization principle, support vector machine, complex random samples, VC dimension