Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons

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AI Summary

This work addresses the prohibitive computational cost that hinders the application of optimal transport to spherical data, such as global climate models. The authors propose a fast Sinkhorn algorithm based on a heat kernel approximation on the sphere, leveraging the structure of spherical harmonics to enable efficient comparison of probability measures with only O(n) memory and O(n^{3/2}) time complexity. Theoretical analysis establishes that the heat kernel–based cost converges to the true optimal transport cost in both balanced and unbalanced settings, while preserving the geometric and analytical properties of the Sinkhorn divergence. Experiments demonstrate substantial computational gains on synthetic data and successful application to spatial and seasonal evaluation of global climate models.

Abstract

Optimal transport provides a powerful framework for comparing measures while respecting the geometry of their support, but comes with an expensive computational cost, hindering its potential application to real world use cases. On manifolds, convolutional algorithms based on the heat kernel have been proposed to alleviate this cost, but their theoretical properties remain largely unexplored. We establish that the heat kernel cost converges to the optimal transport cost as time vanishes in the balanced and unbalanced cases. In the specific case of the 2-sphere $\mathbb{S}^2$, we ensure that the associated Sinkhorn divergences retains the desirable geometric and analytic properties of classical optimal transport discrepancies. Moreover, we leverage the harmonic structure of the sphere to derive a fast Sinkhorn algorithm, requiring only $\mathcal{O}(n)$ memory and $\mathcal{O}(n^{3/2})$ time per iteration, with fully dense GPU-friendly operations. We validate its computational efficiency on synthetic data, and discuss its potential use in the evaluation of global climate models, providing both spatial and seasonal insights into models performances.

Problem

• optimal transport
• spherical harmonics
• computational efficiency
• climate models
• manifold

Innovation

• Spherical Harmonics
• Optimal Transport
• Sinkhorn Algorithm
• Heat Kernel
• Climate Model Evaluation