用球谐函数加速气候模型对比的最优传输计算。
Spherical Harmonic Optimal Transport: Application to Climate Models Comparisons
- 基于球面热核构建快速最优传输算法
- 每轮迭代仅需O(n^{3/2})时间，内存O(n)
- 适合需要地理空间精度的气候模型评估
最优传输为度量比较提供了强大框架，但计算成本高昂。针对流形上的问题，已有基于热核的卷积算法降低开销，但其理论性质尚不明确。本文证明：当时间趋近于零时，热核代价收敛至最优传输代价，涵盖平衡与非平衡情形。在二维球面$\mathbb{S}^2$上，所提出的Sinkhorn散度保持经典最优传输的几何与分析特性。利用球面的调和结构，我们设计了一种高效Sinkhorn算法，每轮迭代仅需$\mathcal{O}(n)$内存和$\mathcal{O}(n^{3/2})$时间，且完全支持GPU密集运算。在合成数据上验证了其计算效率，并探讨其在全局气候模型评估中的应用，可提供空间分布与季节变化的双重洞察。
原文摘要 · Abstract (English)
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.
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