Matched Topological Subspace Detector

Abstract

Topological spaces capture richer relationships than graphs by modeling interactions not only between nodes but also among higher-order entities, such as edges or triangles. This motivates the representation of information defined in irregular domains as topological signals.We focus on simplicial complexes, which are collections of simplices (nodes, edges, triangles and so on) adhering to the inclusion property: a higher-order element cannot exist unless its lower-order components are also part of the complex (e.g., a triangle requires its edges). Accordingly, simplicial signals are functions defined over this set of simplices. By leveraging the spectral dualities of Hodge and Dirac theory, practical simplicial signals often concentrate in specific spectral subspaces (e.g., gradient or curl). For instance, in a foreign currency exchange network, the exchange flow signals typically satisfy the arbitrage-free condition and hence are curl-free. However, the presence of anomalies can disrupt these conditions, causing the signals to deviate from such subspaces. In this work, we formulate a hypothesis testing framework to detect whether simplicial complex signals lie in specific subspaces in a principled and tractable manner. Concretely, we propose Neyman-Pearson matched topological subspace detectors for signals defined at a single simplicial level (such as edges) or jointly across all levels of a simplicial complex. The (energy-based projection) proposed detectors handle missing values, provide closed-form performance analysis, and effectively capture the unique topo-logical properties of the data. We demonstrate the effectiveness of the proposed topological detectors on various real-world data, including foreign currency exchange networks.