Beyond Euclidean Responses:Deep Fréchet Regression for Complex Data Objects

Abstract

Advancements in modern science have led to the increasing availability of non-Euclidean data in metric spaces. Fréchet regression addresses the challenge of modeling relationships between non-Euclidean responses and Euclidean predictors in the absence of linear structure in general metric spaces. Its foundation is the extension of the classical Fréchet mean (Fréchet, 1948) to the conditional Fréchet mean in 2019. Initial implementations included global Fréchet regression, based on a generalization of least squares,  and local Fréchet regression, extending locally weighted smoothing methods to object-valued responses. Early methodological developments further incorporated total variation regularization to accommodate discontinuities and trend-filtering–type behavior, as well as single-index Fréchet regression for multivariate predictors. Deep Fréchet regression addresses scenarios where one encounters large-scale data with high-dimensional predictors and random object responses.  The initial deep Fréchet regression approach (JASA 2025) combines neural networks with low-dimensional object representations obtained through Isomap embeddings. These are coupled with a reverse mapping approach, employing local Fréchet regression to map the low-dimensional manifold representations back to objects in the original metric space. More recent developments have led to fully end-to-end architectures that  currently achieve state-of-the-art performance.
Adapted versions of Fréchet regression have proven instrumental for interpolation and extrapolation of distributions, sliced Wasserstein regression, network regression, global Fréchet manifold learning, and causal inference for random objects.  The presentation will survey the core methodology and emerging machine learning connections.

References: Petersen, A., Müller, H.G. (2019). Fréchet regression for random objects with Euclidean predictors. (Ann. Statist.)
Iao, Su I, Zhou, Y., Müller, H.G. (2025). Deep Fréchet regression (JASA)
Zhou, Y., Iao S.I., Müller, H.G. (2026). End-to-end deep learning for predicting metric space-valued outputs (JMLR)

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