Hodge-Aware Matched Subspace Detectors
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Abstract
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.