Topological Trajectory Classification and Landmark Inference on Simplicial Complexes

TL;DR

This paper introduces a novel method for classifying trajectories on simplicial complexes by learning optimal holes to improve spectral embedding separation, addressing limitations of previous harmonic space approaches.

Contribution

It presents an algorithm to identify key simplices whose removal enhances trajectory classification, extending the method to unsupervised scenarios.

Findings

Effective separation of trajectory classes using learned holes.

Addresses limitations of harmonic eigenspace methods in holeless spaces.

Generalizes approach to unsupervised classification.

Abstract

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.