This paper revisits an adaptation of the random forest algorithm for Fr\'echet regression, addressing the challenge of regression in the context of random objects in metric spaces. Recognizing the limitations of previous approaches, we introduce a new splitting rule that circumvents the computationally expensive operation of Fr\'echet means by substituting with a medoid-based approach. We validate this approach by demonstrating its asymptotic equivalence to Fr\'echet mean-based procedures and establish the consistency of the associated regression estimator. The paper provides a sound theoretical framework and a more efficient computational approach to Fr\'echet regression, broadening its application to non-standard data types and complex use cases.

翻译：本文重新审视了随机森林算法在Fr´echet回归中的改编，旨在解决度量空间中随机对象回归的挑战。鉴于以往方法的局限性，我们引入了一种新的分裂规则，通过采用基于中位数的方法替代计算昂贵的Fr´echet均值操作来规避计算负担。我们通过证明该方法与基于Fr´echet均值的过程渐近等价性来验证其有效性，并建立了相应回归估计量的一致性。本文为Fr´echet回归提供了坚实的理论基础和更高效的计算方法，将其应用范围拓展至非标准数据类型和复杂使用场景。