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.

翻译：我们考虑在由单纯复形建模的离散或离散化二维流形上对轨迹进行分类的问题。先前的研究提出将轨迹投影到霍奇拉普拉斯算子的调和特征空间，然后对得到的嵌入进行聚类。然而，如果所考虑的空间具有零同调（即无“洞”），则1-霍奇拉普拉斯算子的调和空间是平凡的，从而导致该方法失效。本文提出将此问题类比为传感器布局问题，并提出一种旨在学习“最优洞”以区分给定轨迹类别的算法。具体而言，给定一组标记轨迹（我们将其解释为底层单纯复形上的边流），我们搜索其删除能根据轨迹在调和空间中的对应谱嵌入实现轨迹标签最优分离的2-单纯形。最后，我们将此方法推广至无监督场景。