通过构造最优洞来区分轨迹，解决无洞空间分类失效问题。
Topological Trajectory Classification and Landmark Inference on Simplicial Complexes
- 将轨迹视为单纯复形上的边流，通过删减2-单纯形构造可区分的拓扑洞
- 删除特定2-单纯形后，轨迹在调和空间中的谱嵌入能实现标签最优分离
- 适用于无洞空间的有监督与无监督轨迹分类，适合拓扑数据分析者
本文研究在离散或离散化二维流形（以单纯复形建模）上对轨迹进行分类的问题。以往方法将轨迹投影到霍奇拉普拉斯算子的调和特征空间中进行聚类，但当空间同调消失（即无“洞”）时，1-霍奇拉普拉斯算子的调和空间为平凡空间，导致方法失效。为此，我们提出将该问题类比为传感器布局问题，设计一种算法以学习“最优洞”来区分给定的轨迹类别。具体而言，给定一组带标签的轨迹（解释为底层单纯复形上的边流），我们搜索删除某些2-单纯形后，能使轨迹在相应谱嵌入下的调和空间中实现最优标签分离。最后，我们将该方法推广至无监督设置。
原文摘要 · Abstract (English)
We consider the problem of classifying trajectories on a discrete or discretised 2-dimensional manifold modelled by a simplicial complex. Previous works have proposed to project the trajectories into the harmonic eigenspace of the Hodge Laplacian, and then cluster the resulting embeddings. However, if the considered space has vanishing homology (i.e., no "holes"), then the harmonic space of the 1-Hodge Laplacian is trivial and thus the approach fails. Here we propose to view this issue akin to a sensor placement problem and present an algorithm that aims to learn "optimal holes" to distinguish a set of given trajectory classes. Specifically, given a set of labelled trajectories, which we interpret as edge-flows on the underlying simplicial complex, we search for 2-simplices whose deletion results in an optimal separation of the trajectory labels according to the corresponding spectral embedding of the trajectories into the harmonic space. Finally, we generalise this approach to the unsupervised setting.
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