Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons
For climate scientists and practitioners needing to compare spatial data on the sphere, this work provides a computationally efficient optimal transport method with theoretical guarantees.
The authors prove that the heat kernel cost converges to optimal transport cost on manifolds as time vanishes, and develop a fast Sinkhorn algorithm on the sphere with O(n) memory and O(n^{3/2}) time per iteration, enabling efficient comparison of climate models.
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.