提出用于细胞复形的最优传输核，融合结构与特征信息。
Families of Optimal Transport Kernels for Cell Complexes
- 基于霍奇拉普拉斯矩阵推导出细胞复形信号分布的沃尔什距离
- 扩展融合格罗莫夫-沃尔什距离，同时保留特征与拓扑结构信息
- 基于对偶形式构造新核函数，适用于概率测度空间上的学习
近期研究认为细胞复形是理想的机器学习表示形式，但缺乏适用于CW复形的学习方法。本文推导出基于霍奇拉普拉斯矩阵的细胞复形信号分布之间的沃尔什距离显式表达式，从而获得具有结构意义的比较度量和最优传输映射。为同时包含特征与结构信息，将融合格罗莫夫-沃尔什距离推广至CW复形。最后，基于最优传输的对偶形式，引入了定义在CW复形上概率测度空间的新核函数。
原文摘要 · Abstract (English)
Recent advances have discussed cell complexes as ideal learning representations. However, there is a lack of available machine learning methods suitable for learning on CW complexes. In this paper, we derive an explicit expression for the Wasserstein distance between cell complex signal distributions in terms of a Hodge-Laplacian matrix. This leads to a structurally meaningful measure to compare CW complexes and define the optimal transportation map. In order to simultaneously include both feature and structure information, we extend the Fused Gromov-Wasserstein distance to CW complexes. Finally, we introduce novel kernels over the space of probability measures on CW complexes based on the dual formulation of optimal transport.
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