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Statistical analysis is increasingly confronted with complex data from metricspaces. Petersen and M\"uller (2019) established a general paradigm ofFr\'echet regression with complex metric space valued responses and Euclideanpredictors. However, the local approach therein involves nonparametric kernelsmoothing and suffers from the curse of dimensionality. To address this issue,we in this paper propose a novel random forest weighted local Fr\'echetregression paradigm. The main mechanism of our approach relies on a locallyadaptive kernel generated by random forests. Our first method utilizes theseweights as the local average to solve the conditional Fr\'echet mean, while thesecond method performs local linear Fr\'echet regression, both significantlyimproving existing Fr\'echet regression methods. Based on the theory ofinfinite order U-processes and infinite order Mmn -estimator, we establish theconsistency, rate of convergence, and asymptotic normality for our localconstant estimator, which covers the current large sample theory of randomforests with Euclidean responses as a special case. Numerical studies show thesuperiority of our methods with several commonly encountered types of responsessuch as distribution functions, symmetric positive-definite matrices, andsphere data. The practical merits of our proposals are also demonstratedthrough the application to human mortality distribution data and New York taxidata.

Random Forest Weighted Local Fr\'echet Regression with Random Objects