Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. In this paper, we are interested in sketching a set of merge trees. That is, given a set T of merge trees, we would like to find a basis set S such that each tree in T can be approximately reconstructed from a linear combination of merge trees in S. A set of high-dimensional vectors can be sketched via matrix sketching techniques such as principal component analysis and column subset selection. However, up until now, topological descriptors such as merge trees have not been known to be sketchable. We develop a framework for sketching a set of merge trees that combines the Gromov-Wasserstein framework of Chowdhury and Needham with techniques from matrix sketching. We demonstrate the applications of our framework in sketching merge trees that arise from data ensembles in scientific simulations.

中文翻译：

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草绘合并树

合并树是一种拓扑描述符，用于记录标量字段的子级别集之间的连通性。在本文中，我们有兴趣草绘一组合并树。也就是说，给定合并树的集合T，我们希望找到一个基础集S，以便可以根据S中合并树的线性组合来近似重构T中的每个树。可以绘制一组高维向量通过矩阵素描技术，例如主成分分析和列子集选择。但是，到目前为止，还不知道诸如合并树之类的拓扑描述符是可素描的。我们开发了一个草绘一组合并树的框架，该框架将Chowdhury和Needham的Gromov-Wasserstein框架与矩阵草绘中的技术相结合。