Sufficient dimension reduction for regression with metric space-valued responses

TL;DR

This paper introduces a novel dimension reduction technique for regression with responses in general metric spaces, relaxing previous embedding restrictions and demonstrating superior performance on synthetic and real data.

Contribution

It proposes a kernel-free Euclidean embedding method that extends classical dimension reduction techniques to metric space-valued responses.

Findings

Outperforms existing methods on synthetic data

Effectively analyzes COVID-19 transmission factors

Reveals associations between BMI and brain connectivity

Abstract

Data visualization and dimension reduction for regression between a general metric space-valued response and Euclidean predictors is proposed. Current Fr\'ech\'et dimension reduction methods require that the response metric space be continuously embeddable into a Hilbert space, which imposes restriction on the type of metric and kernel choice. We relax this assumption by proposing a Euclidean embedding technique which avoids the use of kernels. Under this framework, classical dimension reduction methods such as ordinary least squares and sliced inverse regression are extended. An extensive simulation experiment demonstrates the superior performance of the proposed method on synthetic data compared to existing methods where applicable. The real data analysis of factors influencing the distribution of COVID-19 transmission in the U.S. and the association between BMI and structural brain connectivity of healthy individuals are also investigated.