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

• URL: http://arxiv.org/abs/2202.10723v1
• Date: Tue, 22 Feb 2022 08:27:58 GMT
• Title: Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics
• Authors: Tam Le and Truyen Nguyen and Dinh Phung and Viet Anh Nguyen
• Abstract summary: 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.
• Score: 25.591913014859184
• License: http://creativecommons.org/licenses/by/4.0/
• 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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