A Dual Method For The Maximal Correlation Problem

The maximal correlation problem (MCP) aiming at assessing the relationship be-tween sets of variables plays a very important role in many areas of statistical applica-tion. Using the method of Lagrange multipliers, the multivariate eigenvalue problem (MEP) is the first order necessary optimality condition for the global solutions of the MCP. Horst method、Gauss-Seidel iteration and P-SOR iteration have been used to solve the MEP, which could help finding the global maximizer of the MCP. Recently, Zhang and Liao proposed an alternating variable method (AVM), which solves the M-CP directly. It even shows a superior performance in the numerical testing, but all the algorithms can not guarantee convergence to a global maximizer of the MCP.First, in this paper, given the dual problem of the MCP, we propose a modified Lagrange dual method, and then analyze the feasibility and validity of the new method. Second, applying the relaxation idea, we analyze the two kinds of inital strategy, which were proposed respectively by Chu and Watterson, Xu. Lastly, we try to present a refined skills to obtain a global solution of MCP using MEP. The numerical results show that it can increase the probability of finding a global maximizer.However,the technique is not applicable to refine the approximate solution produced by AVM,since AVM has the local optimum. So now we present an initial strategy, which is better than that of Zhang and Chu in the numerical experiments.Furthermore, in view of the special case of MCP (m=n), we carry out numeri-cal experiments for the four algorithms, Horst,G-S,P-SOR and AVM. And the result suggests that the convergence rate is faster, but the possibility of convergence to the global solution is little. However, the modified Lagrange dual method can improve the possibility of obtaining a global solution of the MCP.