# Topological Trajectory Classification and Landmark Inference on Simplicial Complexes

arXiv: https://arxiv.org/abs/2412.03145 · HTML: https://mlsanity.com/paper/2412.03145/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 22 references that are in the map.
- First posted on arXiv 4 Dec 2024.

## Builds on (most cited first, 11 of 11)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [The decomposition of the higher-order homology embedding constructed from the k-Laplacian](https://mlsanity.com/paper/2107.10970/) (2021)
- [Representing Edge Flows on Graphs via Sparse Cell Complexes](https://mlsanity.com/paper/2309.01632/) (2023)
- [Disentangling the Spectral Properties of the Hodge Laplacian: Not All Small Eigenvalues Are Equal](https://mlsanity.com/paper/2311.14427/) (2023)
- [Outlier Detection for Trajectories via Flow-embeddings](https://mlsanity.com/paper/2111.13235/) (2021)
- [Point-Level Topological Representation Learning on Point Clouds](https://mlsanity.com/paper/2406.02300/) (2024)
- [Lagrangian Time Series Models for Ocean Surface Drifter Trajectories](https://mlsanity.com/paper/1312.2923/) (2013)

## Built on by (most cited first, 1 of 1)

- [Efficient Sparsification of Simplicial Complexes via Local Densities of States](https://mlsanity.com/paper/2502.07558/) (2025)

## Abstract (arXiv)

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.