Sliced Inverse Regression in Metric Spaces

TL;DR

This paper introduces a nonlinear sufficient dimension reduction method for data in general metric spaces, utilizing reproducing kernel Hilbert spaces and extending sliced inverse regression to non-Euclidean geometries.

Contribution

It develops a novel SDR framework for metric space data by constructing kernel methods based on distance functions, extending classical sliced inverse regression.

Findings

The estimator recovers regression information unbiasedly.

The method is applicable to non-Euclidean data.

Convergence rates are established for the estimator.

Abstract

In this article, we propose a general nonlinear sufficient dimension reduction (SDR) framework when both the predictor and response lie in some general metric spaces. We construct reproducing kernel Hilbert spaces whose kernels are fully determined by the distance functions of the metric spaces, then leverage the inherent structures of these spaces to define a nonlinear SDR framework. We adapt the classical sliced inverse regression of \citet{Li:1991} within this framework for the metric space data. We build the estimator based on the corresponding linear operators, and show it recovers the regression information unbiasedly. We derive the estimator at both the operator level and under a coordinate system, and also establish its convergence rate. We illustrate the proposed method with both synthetic and real datasets exhibiting non-Euclidean geometry.