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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.17072 |
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| _version_ | 1866914103563386880 |
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| author | Kim, Kyum Chen, Yaqing Dubey, Paromita |
| author_facet | Kim, Kyum Chen, Yaqing Dubey, Paromita |
| contents | 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 leverages 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. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17072 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses Kim, Kyum Chen, Yaqing Dubey, Paromita Machine Learning Methodology 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 leverages 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. 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. |
| title | DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses |
| topic | Machine Learning Methodology |
| url | https://arxiv.org/abs/2510.17072 |