DFNN: A Deep Frechet Neural Network Framework for Learning Metric-Space-Valued Responses

arXiv:2510.17072v2 Announce Type: replace-cross Abstract: Regression with non-Euclidean responses---e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions---has become increasingly important in modern applications. In this paper, we propose deep Frechet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses---which are considered as random objects in a metric space---from Euclidean predictors. Our method utilizes the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Frechet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Frechet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses, without making model assumptions or relying on local smoothing. We further establish rigorous generalization guarantees for DFNNs and derive corresponding risk bounds, providing, to the best of our knowledge, the first such theoretical results for deep learning regression with metric-space-valued responses. Empirical studies on synthetic distributional and network-valued responses, as well as real-world applications to predicting compositional responses in an Aitchison simplex and spherical responses, demonstrate that DFNNs consistently outperform all existing methods.

Source: arXiv cs.LG | 2026-09-10