Deep Fr\'echet Regression

TL;DR

This paper introduces a deep neural network-based regression model for non-Euclidean responses in metric spaces, combining manifold learning and local Fréchet regression to handle high-dimensional predictors and complex data structures.

Contribution

It proposes a novel deep Fréchet regression framework that integrates manifold learning and local Fréchet regression, addressing high-dimensionality and nonlinearity in non-Euclidean data.

Findings

Model outperforms existing methods in simulations

Effective handling of high-dimensional predictors

Theoretical convergence guarantees provided

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 Fr\'echet 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 Fr\'echet 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.