Deep Fréchet Regression
Su I Iao, Yidong Zhou, Hans-Georg Müller
TL;DR
This paper develops Deep Fréchet Regression (DFR) for regressing metric-space–valued responses on high-dimensional Euclidean predictors. The framework combines deep neural networks to learn low-dimensional manifold representations with local Fréchet regression to map back to the original metric space, using ISOMAP for manifold estimation and an errors-in-variables treatment. The authors establish convergence rates for the DNN component under dependent sub-Gaussian noise with bias, extend local Fréchet regression to multivariate predictors with predictor errors, and derive the overall rate for the DFR estimator. They demonstrate by simulations and real-data applications (NYC taxi networks and age-at-death distributions) that DFR outperforms existing methods, with robustness to manifold approximation and practical interpretability. The work contributes a versatile, scalable approach for non-Euclidean responses, combining neural and geometric tools with rigorous statistical theory.
Abstract
Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This paper addresses the challenge of modeling relationships between non-Euclidean responses and multivariate Euclidean predictors. We propose a flexible regression model capable of handling high-dimensional predictors without imposing parametric assumptions. Two primary challenges are addressed: the curse of dimensionality in nonparametric regression and the absence of linear structure in general metric spaces. The former is tackled using deep neural networks, while for the latter we demonstrate the feasibility of mapping the metric space where responses reside to a low-dimensional Euclidean space using manifold learning. We introduce a reverse mapping approach, employing local Fréchet regression, to map the low-dimensional manifold representations back to objects in the original metric space. We develop a theoretical framework, investigating the convergence rate of deep neural networks under dependent sub-Gaussian noise with bias. The convergence rate of the proposed regression model is then obtained by expanding the scope of local Fréchet regression to accommodate multivariate predictors in the presence of errors in predictors. Simulations and case studies show that the proposed model outperforms existing methods for non-Euclidean responses, focusing on the special cases of probability distributions and networks.