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运筹学学报

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半监督度量学习内蕴最速下降算法的收敛性分析

李鑫1,2   白延琴3,*   

  1. 1. 上海大学经济学院, 上海 200444 2. 南阳师范学院数学与统计学院, 河南南阳 473061 3. 上海大学理学院数学系, 上海  200444
  • 收稿日期:2017-01-10 出版日期:2017-09-15 发布日期:2017-09-15
  • 通讯作者: 白延琴 yqbai@t.shu.edu.cn
  • 基金资助:

    国家自然科学基金(Nos.11371242, 11471208, 11131006, 11101260)

Convergence analysis of an intrinsic steepest descent method on semi-supervised metric learning

LI Xin1,2  BAI Yanqin3,*   

  1. 1. School of Economics, Shanghai University, Shanghai 200444, China 2. School of Mathematics and Statistics, Nanyang Normal University, Nangang 473061, Henan, China 3. Department of Mathematics, College of Science, Shanghai University, Shanghai 200444, China
  • Received:2017-01-10 Online:2017-09-15 Published:2017-09-15

PDF

504

被引次数

1

摘要：

主要研究对称正定矩阵群上的内蕴最速下降算法的收敛性问题. 首先针对一个可转化为对称正定矩阵群上无约束优化问题的半监督度量学习模型, 提出对称正定矩阵群上一种自适应变步长的内蕴最速下降算法. 然后利用李群上的光滑函数在任意一点处带积分余项的泰勒展开式, 证明所提算法在对称正定矩阵群上是线性收敛的. 最后通过在分类问题中的数值实验说明算法的有效性.

关键词: 度量学习, 内蕴最速下降算法, 对称正定矩阵群, 李群

Abstract:

In this paper, we derive the convergence problem of an intrinsic steepest descent algorithm for semi-supervised metric learning problem on symmetric positive definite matrices groups.We first rewrite semi-supervised metric learning problem into an unconstrained optimization problem on symmetric positive definite matrices groups. Then we present an intrinsic steepest descent algorithm with an adaptive iteration step-size. Moreover, we prove that the algorithm converges linearly by using a Taylor's expansion of smooth function at any point in Lie groups. Finally, we show a few numerical experiments on classification problem to demonstrate the effectiveness of the proposed algorithm.

Key words: metric learning, intrinsic steepest descent algorithm, symmetric positive definite matrices groups, Lie groups