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.

翻译：我们提出一种方法来检测实验观察到的离散或离散的多式外向轨迹的外向。 我们的方法类似于光谱嵌入器, 如扩散映射和 Laplacian egenmaps, 用于从与低电子值相关的图 Laplacian 的叶形嵌入螺旋体。 我们在这里将轨迹视为由简单可解释的、 低度的合成法和典型的合成法分析的轨迹。 我们展示了这种单轨迹相对于复合的同理学的行为, 这与深层空间的孔相对应。 这使我们能够根据简单可解释的、 低度的合成法分析轨迹对轨迹进行分类, 从而得出这些边缘流的嵌入。 我们通过投射轨迹向Hodge 1- Laplicacian 与小电子值相联系的叶形体外空间。 我们展示了这种单轨迹相对于复合法的行为, 与基础空间的洞洞相对。 这让我们能够根据简单的、 低度的合成法和典型的合成法学方法的演化方法, 展示了我们如何将这种单轨迹从不同的实验性向外演化。