Title: End-to-End Deep Learning for Predicting Metric Space-Valued Outputs

Abstract

A wide range of contemporary applications requires the prediction of structured outputs that reside in non-Euclidean domains, including symmetric positive-definite matrices, networks, and probability distributions. Because these outputs are inherently represented as elements of general metric spaces, traditional regression methods—which depend on vector space structures—become inapplicable. To address this challenge, we present E2M (End-to-End Metric regression), a novel deep learning framework designed specifically for predicting metric space-valued targets.

In this approach, predictions are generated by calculating weighted Fréchet means of the training outputs. The weights assigned to these outputs are determined by a neural network that is conditioned on the input data. This methodology offers a theoretically sound mechanism for geometry-aware prediction. It eliminates the need for surrogate embeddings and restrictive parametric assumptions, thereby maintaining the intrinsic geometry of the output space intact.

We provide robust theoretical foundations for the framework, including a universal approximation theorem that defines the model’s expressive power and a convergence analysis of the entropy-regularized training objective. Our extensive simulations, which cover probability distributions, networks, and symmetric positive-definite matrices, demonstrate that E2M consistently delivers state-of-the-art results. Notably, the model’s performance advantages become increasingly significant as sample sizes grow. Furthermore, real-world applications involving human mortality distributions and New York City taxi networks highlight the framework’s practical utility and adaptability.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC