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

Researchers introduce E2M (End-to-End Metric regression), a deep learning framework that predicts non-Euclidean outputs like probability distributions and networks by computing weighted Fréchet means with neural network-learned weights. The method preserves geometric properties of output spaces while achieving state-of-the-art performance across multiple domains without requiring surrogate embeddings.

E2M addresses a fundamental limitation in machine learning: most regression techniques assume vector space structure, making them unsuitable for predicting outputs that live in non-Euclidean spaces. This research bridges that gap by leveraging Fréchet means—a generalization of averaging to metric spaces—combined with neural network-based weight learning. The framework's theoretical foundation includes universal approximation guarantees and convergence analysis, establishing rigor alongside practical utility.

The problem itself emerges from real-world applications where outputs naturally inhabit geometric spaces. Predicting probability distributions, network topologies, or symmetric positive-definite matrices traditionally requires either lossy embeddings into Euclidean space or domain-specific parametric models that impose restrictive assumptions. E2M circumvents these limitations by respecting intrinsic geometry throughout the prediction pipeline.

For the AI and machine learning community, this work has broad applicability. Fields ranging from statistical modeling to graph prediction to matrix-valued estimation could benefit from geometry-preserving regression. The empirical validation spans synthetic benchmarks and real applications—mortality distribution prediction and NYC taxi network analysis—demonstrating practical relevance beyond theoretical novelty.

Looking forward, the framework's scalability and integration with modern deep learning infrastructure will determine adoption. Questions about computational efficiency for high-dimensional metric spaces and performance with limited training data warrant investigation. The work opens avenues for geometry-aware prediction across domains previously constrained by Euclidean-space assumptions.

• →E2M performs regression on metric space-valued outputs using weighted Fréchet means with learned neural network weights
• →The framework preserves intrinsic geometry without surrogate embeddings or restrictive parametric assumptions
• →Universal approximation theorem and convergence guarantees establish theoretical rigor for the approach
• →Empirical results demonstrate state-of-the-art performance on probability distributions, networks, and symmetric matrices
• →Applications span mortality prediction and transportation networks, showing practical utility across domains