求解多组变量典型相关分析Maxrat准则的预处理Dinkelbach方法

Preconditioned Dinkelbach Methods for Solving the Maxrat Criterion of Multiple-sets Canonical Correlation Analysis

多组变量典型相关分析的Maxrat准则是一类具约束的非线性最优化问题.本文给出了关于最优性的一阶必要条件和一个便于应用的充分条件.利用Dinkelbach技巧给出了求解Maxrat的一种算法.提出了几种初始点策略用于改进算法的收敛速度和提高收敛到全局最优解的可能性.数值实验结果证明算法和初始点策略是有效的.

Canonical correlation analysis （CCA） aims at assessing the relationship between sets of random variables and plays an important role in many areas of statistical applications. This paper deals with numerical methods for the Maxrat criterion of multiple-sets canonical correlation analysis. Mathematically, Maxrat is a constrained nonlinear optimization problem and solving it globally still remains very challenging. Towards the global solution of Maxrat, we have obtained several results in the present paper. Optimality conditions are derived. Upper and lower bounds of the optimal objective function value are presented. A Dinkelbach method is proposed and analysed. Several starting point strategies are suggested to improve the iterative method in both reducing the number of iterations and boosting up the probability of finding a global solution of Maxrat. Numerical results are presented to demonstrate the efficiency of these algorithms and the starting point strategies.