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This paper revisits an adaptation of the random forest algorithm forFr\'echet regression, addressing the challenge of regression in the context ofrandom objects in metric spaces. Recognizing the limitations of previousapproaches, we introduce a new splitting rule that circumvents thecomputationally expensive operation of Fr\'echet means by substituting with amedoid-based approach. We validate this approach by demonstrating itsasymptotic equivalence to Fr\'echet mean-based procedures and establish theconsistency of the associated regression estimator. The paper provides a soundtheoretical framework and a more efficient computational approach to Fr\'echetregression, broadening its application to non-standard data types and complexuse cases.

Medoid splits for efficient random forests in metric spaces