Outlier Detection for Trajectories via Flow-embeddings

TL;DR

This paper introduces a novel spectral embedding method for detecting outliers in trajectory data on complex manifolds using Hodge Laplacians, enabling topological classification of trajectories.

Contribution

It develops a new outlier detection technique based on Hodge 1-Laplacian embeddings of edge-flows on simplicial complexes, extending spectral methods to higher-order structures.

Findings

Effective outlier detection demonstrated on synthetic data

Successfully applied to empirical trajectory data

Classifies trajectories based on topological differences

Abstract

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.