DFNN: A Deep Fréchet Neural Network Framework for Learning Metric-Space-Valued Responses
This addresses the problem of predicting complex, non-Euclidean data types in applications like occupational composition analysis, representing an incremental advancement in neural network frameworks for metric-space-valued responses.
The paper tackles regression with non-Euclidean responses, such as probability distributions and networks, by proposing deep Fréchet neural networks (DFNNs) to predict these responses from Euclidean predictors, and demonstrates that DFNNs consistently outperform existing methods in empirical studies.
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.