aiqbench analysis
This research introduces Deep Fréchet Neural Networks (DFNNs), a novel framework designed to handle regression tasks where outputs are non-Euclidean objects like probability distributions or networks rather than standard vectors. By establishing universal approximation theorems and rigorous generalization guarantees for metric-space-valued responses, the work significantly advances theoretical understanding of deep learning beyond traditional Euclidean spaces. The approach demonstrates superior performance in predicting complex data structures such as compositional responses and spherical data without relying on local smoothing assumptions.
DFNN: A Deep Fr\'echet 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 Fr\'echet 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 Fr\'echet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fr\'echet 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.
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 Fréchet 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 Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet 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.
Machine Learning (stat.ML); Machine Learning (cs.LG); Methodology (stat.ME)
arXiv:2510.17072 [stat.ML]
(or arXiv:2510.17072v2 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2510.17072 Focus to learn more arXiv-issued DOI via DataCite
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