# Outlier Detection for Trajectories via Flow-embeddings

arXiv: https://arxiv.org/abs/2111.13235 · HTML: https://mlsanity.com/paper/2111.13235/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 20 references that are in the map.
- First posted on arXiv 25 Nov 2021.

## Builds on (most cited first, 8 of 8)

- [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)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Signal Processing on Cell Complexes](https://mlsanity.com/paper/2110.05614/) (2021)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)

## Built on by (most cited first, 4 of 4)

- [Topological Point Cloud Clustering](https://mlsanity.com/paper/2303.16716/) (2023)
- [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)
- [Topological Trajectory Classification and Landmark Inference on Simplicial Complexes](https://mlsanity.com/paper/2412.03145/) (2024)

## Abstract (arXiv)

We propose a method to detect outliers in empirically observed trajectories on a discrete or discretized manifold modeled by a simplicial complex. Our approach is similar to spectral embeddings such as diffusion-maps and Laplacian eigenmaps, that construct vertex embeddings from the eigenvectors of the graph Laplacian associated with low eigenvalues. Here we consider trajectories as edge-flow vectors defined on a simplicial complex, a higher-order generalization of graphs, and use the Hodge 1-Laplacian of the simplicial complex to derive embeddings of these edge-flows. By projecting trajectory vectors onto the eigenspace of the Hodge 1-Laplacian associated to small eigenvalues, we can characterize the behavior of the trajectories relative to the homology of the complex, which corresponds to holes in the underlying space. This enables us to classify trajectories based on simply interpretable, low-dimensional statistics. We show how this technique can single out trajectories that behave (topologically) different compared to typical trajectories, and illustrate the performance of our approach with both synthetic and empirical data.