Fréchet Sufficient Dimension Reduction for Random Objects

We in this paper consider Fr\échet sufficient dimension reduction with responses being complex random objects in a metric space and high dimension Euclidean predictors. We propose a novel approach called weighted inverse regression ensemble method for linear Fr\échet sufficient dimension reduction. The method is further generalized as a new operator defined on reproducing kernel Hilbert spaces for nonlinear Fr\échet sufficient dimension reduction. We provide theoretical guarantees for the new method via asymptotic analysis. Intensive simulation studies verify the performance of our proposals. And we apply our methods to analyze the handwritten digits data to demonstrate its use in real applications.