Deep Fréchet Regression

Su I Iao, Yidong Zhou and Hans-Georg Müller

Journal of the American Statistical Association, 2025, vol. 120, issue 551, 1437-1448

Abstract: Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. This article 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. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

Date: 2025
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DOI: 10.1080/01621459.2025.2507982

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