On The Sample Complexity of HGR Maximal Correlation Functions

Shao-Lun Huang, Xiangxiang Xu

Abstract: The Hirschfeld-Gebelein-Rényi (HGR) maximal correlation has been shown useful in many machine learning scenarios. In this paper, we investigate the sample complexity problem of estimating the HGR maximal correlation functions by the alternative conditional expectation (ACE) algorithm from a sequence of training data in the asymptotic regime. Specifically, we develop a mathematical framework to characterize the eigen-decomposition of perturbed matrices, and then establish the error exponent of the learning error for the computed HGR maximal correlation functions. Our result essentially indicates the number of training samples required for estimating the HGR maximal correlation functions to a targeted accuracy by the ACE algorithm.