Generative Modeling on Metric Graphs via Neural Optimal Transport

2026-06-16 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

A novel deep generative modeling framework has been introduced for probability distributions continuously supported on compact metric graphs. This method embeds the graph into a smooth ambient space, then solves an entropic Kantorovich problem using a neural semidual parameterization, and finally projects generated samples back onto the original graph. The framework explores two embedded geometries: an extrinsic Euclidean realization and the intrinsic tropical Abel–Jacobi embedding into the Jacobian torus. Crucially, the resulting generator is graph-supported by construction. Theoretical guarantees show that, with increasing neural expressivity and vanishing heat scale, the learned generator weakly converges to a valid transport coupling. Empirically, the method outperforms heuristic baselines based on discrete graph Optimal Transport across various metric graphs and demonstrates scalability on one million Uber pickup locations in Manhattan, New York City.

Key takeaway

For Machine Learning Engineers developing generative models for network-constrained data, this framework offers a robust solution. You should consider implementing this neural optimal transport approach, especially for large-scale datasets like urban mobility, as it demonstrably improves accuracy over discrete graph OT baselines and scales more favorably. This allows you to generate continuous, graph-supported samples that respect the underlying network geometry, avoiding mesh-dependent approximations.

Key insights

A neural optimal transport framework enables continuous generative modeling on metric graphs by embedding, solving, and projecting.

Principles

• Graph-supported generative models must respect metric geometry.
• Neural Optimal Transport scales for high-dimensional continuous domains.
• Embedding graphs into smooth spaces enables neural OT application.

Method

The method embeds a metric graph into a smooth ambient space, solves an entropic Kantorovich problem via neural semidual parameterization, and projects heat-smoothed samples back onto the original graph to ensure graph-supported outputs.

In practice

• Model urban mobility data, e.g., Uber pickup locations.
• Benchmark generative models on diverse metric graph geometries.

Topics

• Generative Modeling
• Metric Graphs
• Neural Optimal Transport
• Graph Embeddings
• Urban Mobility
• Optimal Transport

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.