arXiv:2509.23544v2 Announce Type: replace-cross Abstract: Many modern applications involve predicting structured, non-Euclidean outputs such as probability distributions, networks, and symmetric positive-definite matrices. These outputs are naturally modeled as elements of general metric spaces, where classical regression techniques that rely on vector space structure no longer apply. We introduce E2M (End-to-End Metric regression), a deep learning framework for predicting metric space-valued outputs. E2M performs prediction via weighted Fr\'echet means over training outputs, where the weights
The increasing complexity and diversity of AI applications demand new mathematical frameworks to handle non-Euclidean data types, pushing research towards more generalizable learning models.
This development allows AI to effectively process and predict outputs in complex, non-standard data spaces, opening up new domains for AI automation and analysis where classical techniques failed.
AI models will no longer be restricted to Euclidean spaces, enabling direct and more accurate prediction of structured outputs like graphs, distributions, and matrices in an end-to-end fashion.
- · AI researchers and deep learning practitioners
- · Bioinformatics and materials science sectors
- · Financial modeling and risk analysis
- · Autonomous systems developers
- · Classical statistical methods in specialized domains
- · Organizations reliant on manual feature engineering for complex data structures
More accurate and efficient AI models for highly structured data become feasible across various applications.
This framework could accelerate advancements in fields requiring complex output modeling, such as drug discovery, climate modeling, or network optimization.
The enhanced capability for AI to understand and predict non-Euclidean structures may contribute to the development of more sophisticated AI agents capable of higher-level reasoning and interaction within complex environments.
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Read at arXiv cs.LG