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arXiv · 2202.10723

Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics

Tam Le·Truyen Nguyen·Dinh Phung·Viet Anh Nguyen
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Abstract

Optimal transport (OT) is a popular measure to compare probability distributions. However, OT suffers a few drawbacks such as (i) a high complexity for computation, (ii) indefiniteness which limits its applicability to kernel machines. In this work, we consider probability measures supported on a graph metric space and propose a novel Sobolev transport metric. We show that the Sobolev transport metric yields a closed-form formula for fast computation and it is negative definite. We show that the space of probability measures endowed with this transport distance is isometric to a bounded convex set in a Euclidean space with a weighted $\ell_p$ distance. We further exploit the negative definiteness of the Sobolev transport to design positive-definite kernels, and evaluate their performances against other baselines in document classification with word embeddings and in topological data analysis.

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Tam Le, Truyen Nguyen, Dinh Phung, Viet Anh Nguyen. 2022-02-22. Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics. https://arxiv.org/abs/2202.10723

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