  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.
