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

TL;DR

This work extends regression to metric-space-valued responses by introducing DFNN, an end-to-end deep network whose output is a weighted Fréchet mean, guaranteeing predictions stay in the response space. A Fréchet-risk objective guides training, and a universal approximation theorem ensures DFNN can approximate the conditional Fréchet mean arbitrarily well under mild regularity. Empirical results across Wasserstein, network-graph, and compositional spaces show DFNN outperforms global Fréchet regression, SDR, DFR, and standard DNN baselines, demonstrating robust predictive performance for non-Euclidean responses. The framework generalizes standard DNNs to non-Euclidean targets, with potential impact on domains requiring distributional, network-valued, or compositional predictions from high-dimensional predictors.

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 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.

Figures (3)

• Figure 1: Boxplots of MSPEs for GFR, SDR, DFR, and DFNN for Experiment \ref{['case:distn_lowdim2']} with sample size $n\in\{200,500,1000\}$.
• Figure 2: Boxplots of MSPEs for GFR, SDR, DFR, and DFNN for Experiment \ref{['case:network']} with sample size $n\in\{200,500,1000\}$ and different levels of noise ($a=0$ corresponding to noiseless settings and larger $a$ corresponding to higher levels of noise). Three outliers with MSPE values above 10, all from the DFR method, occurred under the settings ($n=200, a=0.02$), ($n=1000, a=0.02$), and ($n=1000, a=0.1$), and are omitted for visualization purposes.
• Figure 3: Boxplots of MSPEs from $100$ times $10$-fold cross-validation for GFR, SDR, DFR, and DFNN for Experiment \ref{['case:employment_composition']}.

Theorems & Definitions (2)

• Remark 1
• Theorem 1