On the Asymptotic HGR Maximal Correlation of Gaussian Markov Chain

The Hirschfeld-Gebelein-Renyi (HGR) maximal correlation shows widespread applications in statistics and machine learning fields. This paper explores the HGR maximal correlation among two discrete time random processes that form Markov chains with infinite chain lengths. Under the specific form of Gaussian random variables, the optimal correlation functions are linear to the data. Therefore, this problem can be reduced to solving the largest singular value of a particular matrix. Then, we present the analytical expression of the asymptotic HGR maximal correlation, where a geometric interpretation is also provided. This study offers insights into the effective design of feature extraction in machine learning tasks.