A0320
Title: Dimension reduction and data visualization for Frechet regression
Authors: Qi Zhang - The Pennsylvania State University (United States)
Lingzhou Xue - Penn State University (United States) [presenting]
Bing Li - The Pennsylvania State University (United States)
Abstract: Frechet regression models provide a promising framework for regression analysis with metric space-valued responses. We introduce a flexible sufficient dimension reduction (SDR) method for Frechet regression to achieve two purposes: to mitigate the curse of dimensionality caused by high-dimensional predictors, and to provide a tool for data visualization for Frechet regression. The approach is flexible enough to turn any existing SDR method for Euclidean $(X,Y)$ into one for Euclidean $X$ and metric space-valued $Y$. The basic idea is to first map the metric-space valued random object $Y$ to a real-valued random variable $f(Y)$ using a class of functions, and then perform classical SDR to the transformed data. If the class of functions is sufficiently rich, then we are guaranteed to uncover the Frechet SDR space. We showed that such a class, which we call an ensemble, can be generated by a universal kernel. We established the consistency and asymptotic convergence rate of the proposed methods. The finite-sample performance of the proposed methods is illustrated through simulation studies for several commonly encountered metric spaces that include Wasserstein space, the space of symmetric positive definite matrices, and the sphere. We illustrated the data visualization aspect of our method in real applications.
