The Fr\'echet correlation coefficient for heterogeneous random objects

Modern regression analysis often involves responses and predictors taking values in the same or distinct metric spaces. To rank non-Euclidean heterogeneous predictors in regression by explanatory strength, analogous to the classical $R^2$, we introduce the Fr\'echet correlation coefficient (FCC), defined as the relative reduction in the Fr\'echet variance of the response after conditioning on a specific predictor. FCC is directional, model-free, and interpretable on a unit-scale, attaining one under almost sure functional dependence and zero when the Fr\'echet mean is invariant to conditioning. We propose a novel partition-based estimator that avoids explicit nonparametric estimation of the conditional Fr\'echet mean function, thereby improving both computational efficiency and flexibility. A tailored wild bootstrap algorithm is further developed for testing the Fr\'echet conditional mean dependence. We establish asymptotic theory and evaluate power through extensive simulations.