Towards the global solution of the maximal correlation problem

The maximal correlation problem (MCP) aiming at optimizing correlation between sets of variables plays a very important role in many areas of statistical applications. Currently, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP) for a related matrix A. The MEP is a necessary condition for the global solutions of the MCP. Up to date, there is no algorithm that can guarantee convergence to a global maximizer of the MCP, which would have significant impact in applications. Towards the global solutions of the MCP, we have obtained four results in the present paper. First, the sufficient and necessary condition for global optimality of the MCP when A is a positive matrix is extended to include A being a nonnegative irreducible matrix. Secondly, the uniqueness of the multivariate eigenvalues in the global maxima of the MCP is proved either when there are only two sets of variables involved, or when A is nonnegative irreducible, regardless of the number of sets of variables. The uniqueness of the global solution of the MCP for the latter case is also proved. These theoretical achievements lead to our third result that either the Horst-Jacobi algorithm or the Gauss-Seidel algorithm converges globally to a global maximizer of the MCP as long as A is a nonnegative irreducible matrix. Lastly, some new estimates of the multivariate eigenvalues related to the global maxima are obtained.