Topological Trajectory Classification and Landmark Inference on Simplicial Complexes

TL;DR

This paper presents a topology-aware framework for trajectory classification on simplicial complexes by learning a set of landmark holes whose removal yields harmonic embeddings that best separate trajectory classes. By constraining holes to single 2-simplices and computing harmonic vectors through sparse least-squares with $B_2$, the authors enable scalable embedding of edge-flows without full eigen decompositions. A local-search strategy combined with diffusion of edge-flows produces robust, low-dimensional representations suitable for supervised and unsupervised clustering on large networks (e.g., $\sim 10^5$ nodes). Demonstrations on synthetic data and ocean-drifter datasets show effective landmark discovery and accurate trajectory discrimination, including unsupervised discovery of coastal landmarks.

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.

Figures (5)

• Figure 1: Sample run of the proposed method on a simplicial complex with $3$ holes and $5$ trajectory classes.Top row, left: Sample trajectories in the training set, where large dots represent the start of the trajectories. Centre left: Goal function projected down to simplices, highest possible value shown. Yellow values indicate higher score. We can see that the regions of good cluster scores represent the areas between trajectory classes, being good candidates for landmarks chosen by the method. Centre right: Achieved ARI depending on number of holes used for classification. We see that $2$ holes already yield optimal classification performance in this run. Right: Harmonic flow corresponding to the top chosen $2$-simplex/hole. The strength of the harmonic flow is strongest at the boundary of the hole and then decreases with increasing distance. Bottom left: Landmarks selected by algorithm, corresponding to simplices which result in good cluster score. Centre left: Increase of the target function throughout optimisation run. Centre right and right: Harmonic flows corresponding to remaining two selected landmarks/holes.
• Figure 2: Diffusion processes on edge flowsLeft: Trajectory under diffusion for small time step $\tau$. Right: Trajectory under diffusion for larger time step $\tau$. We do not show the underlying network.
• Figure 3: Performance in synthetic experiments. We set $n_\text{train}=5n_\text{classes}$ and $n_\text{test}=50n_\text{classes}$, showing that our proposed method works even for few trajectory examples. We vary the number of total classes and report the adjusted rand index (ARI, left, takes values in $[-0.5,1]$, $1$ for perfect recovery, $0$ for random cluster assignments.) and time (only for the supervised case, right). Because we keep the size of the network fixed, more classes not only require a more fine grained classifier, but they also are more likely to overlap and thus lead to a harder problem. $n_\text{holes}$ indicate the number of holes/landmarks the method is allowed to introduce to classify the trajectories.
• Figure 4: Unsupervised trajectory classification on the Ocean Drifter dataset. We overlay the goal function of the simplices (red) with the two trajectory classes (blue), and the grid (black). The method is able to capture a landmark corresponding to the north-west coast of Madagascar. Furthermore, it produces two clusters corresponding to drifters passing to the north and to the west of Madagascar, without having access to any labelled data.
• Figure : Proposed topology inference and trajectory classification method.