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

• URL: http://arxiv.org/abs/2510.17072v1
• Date: Mon, 20 Oct 2025 00:57:30 GMT
• Title: DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses
• Authors: Kyum Kim, Yaqing Chen, Paromita Dubey,
• Abstract summary: deep Fr'echet neural networks (DFNNs) are an end-to-end deep learning framework for predicting non-Euclidean responses from Euclidean predictors.<n>We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory.<n> Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.
• Score: 0.25778694761493826
• License: http://creativecommons.org/licenses/by/4.0/
• 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 leverages 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. Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.

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