Statistica Sinica 32 (2022), 2315-2337

SLICED INVERSE REGRESSION IN METRIC SPACES

Joni Virta, Kuang-Yao Lee and Lexin Li

University of Turku, Temple University and University of California at Berkeley

Abstract: In this article, we propose a general nonlinear sufficient dimension reduction(SDR) framework when both the predictor and the response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces with kernels that are fully determined by the distance functions of the metric spaces, and then leverage the inherent structures of these spaces to define a nonlinear SDR framework. We adapt the classical sliced inverse regression within this framework for the metric space data. Next we build an estimator based on the corresponding linear operators, and show that it recovers the regression information in an unbiased manner. We derive the estimator at both the operator level and under a coordinate system, and establish its convergence rate. Lastly, we illustrate the proposed method using synthetic and real data sets that exhibit non-Euclidean geometry.

Key words and phrases: Covariance operator, metric space, reproducing kernel Hilbert space, sliced inverse regression, sufficient dimension reduction.