TY - JOUR
T1 - An alternating variable method for the maximal correlation problem
AU - Zhang, Lei Hong
AU - Liao, Li Zhi
N1 - Funding Information:
This research was supported in part by FRG grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong.
PY - 2012/9
Y1 - 2012/9
N2 - The maximal correlation problem (MCP) aiming at optimizing correlations between sets of variables plays an important role in many areas of statistical applications. Up to date, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP), which serves only as a necessary condition for the global maxima of the MCP. For statistical applications, the global maximizer is quite desirable. In searching the global solution of the MCP, in this paper, we propose an alternating variable method (AVM), which contains a core engine in seeking a global maximizer. We prove that (i) the algorithm converges globally and monotonically to a solution of the MEP, (ii) any convergent point satisfies a global optimal condition of the MCP, and (iii) whenever the involved matrix A is nonnegative irreducible, it converges globally to the global maximizer. These properties imply that the AVM is an effective approach to obtain a global maximizer of the MCP. Numerical testings are carried out and suggest a superior performance to the others, especially in finding a global solution of the MCP.
AB - The maximal correlation problem (MCP) aiming at optimizing correlations between sets of variables plays an important role in many areas of statistical applications. Up to date, algorithms for the general MCP stop at solutions of the multivariate eigenvalue problem (MEP), which serves only as a necessary condition for the global maxima of the MCP. For statistical applications, the global maximizer is quite desirable. In searching the global solution of the MCP, in this paper, we propose an alternating variable method (AVM), which contains a core engine in seeking a global maximizer. We prove that (i) the algorithm converges globally and monotonically to a solution of the MEP, (ii) any convergent point satisfies a global optimal condition of the MCP, and (iii) whenever the involved matrix A is nonnegative irreducible, it converges globally to the global maximizer. These properties imply that the AVM is an effective approach to obtain a global maximizer of the MCP. Numerical testings are carried out and suggest a superior performance to the others, especially in finding a global solution of the MCP.
KW - Canonical correlation
KW - Gauss-Seidal method
KW - Global maximizer
KW - Maximal correlation problem
KW - Multivariate eigenvalue problem
KW - Multivariate statistics
KW - Power method
UR - http://www.scopus.com/inward/record.url?scp=84865129768&partnerID=8YFLogxK
U2 - 10.1007/s10898-011-9758-2
DO - 10.1007/s10898-011-9758-2
M3 - Journal article
AN - SCOPUS:84865129768
SN - 0925-5001
VL - 54
SP - 199
EP - 218
JO - Journal of Global Optimization
JF - Journal of Global Optimization
IS - 1
ER -