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.

Matched subspace detection (MSD) is a powerful tool recently generalized from Euclidean data to graph signal processing. However, existing graph-based MSD methods are often limited by assumptions of known noise variance and by overlooking the statistical properties of the graph Fourier transform (GFT) coefficients thereby limiting practical applicability. To address these gaps, this paper introduces two novel generalized likelihood ratio (GLR) tests for graph-based MSD. The first-order GLR test operates without knowledge of the noise variance and the GFT coefficients by estimating them via maximum likelihood. The second-order GLR test further incorporates a Gaussian prior on the GFT coefficients, yielding a more powerful and comprehensive statistical model. Experimental results demonstrate that our proposed detectors are robust and effective, particularly in challenging noisy scenarios, highlighting their importance for detection tasks in graph signal processing.

This paper introduces a hypothesis testing problem to detect whether a noisy simplicial signal lives in some specific Hodge subspaces or not. This is of particular relevance for edge flows in a network since they exhibit, under normal circumstances, different properties in Hodge decomposition. For example, a traffic flow in a road network is often conservative and that can be localized in a particular Hodge subspace. We propose two Neyman-Pearson optimal detectors for this task: the Simplicial Hodge Detector (SHD) and the Constrained Simplicial Hodge Detector (CSHD). They compare the energy of the simplicial embeddings in different Hodge subspaces and distinguish between the two hypotheses. The SHD utilizes the maximum likelihood estimation, while CSHD incorporates signal prior information to estimate the simplicial embeddings. These detectors are validated through numerical simulations on both real-world and synthetic data, indicating great potential in practical applications.