Matched Topological Subspace Detector

2025-04-08

AI Summary

This work addresses anomaly detection in topological signals defined on simplicial complexes, focusing on deviations arising from structural violations—such as breaches of irrotationality or solenoidality constraints imposed by Hodge/Dirac spectral subspaces. We propose a Neyman–Pearson-optimal matched subspace detection framework, the first to support both single-layer (e.g., edge signals) and multi-layer joint topological subspace testing. The method is robust to missing values, admits closed-form performance analysis, and explicitly incorporates higher-order topological priors encoded in the simplicial complex. Leveraging spectral theory of the Hodge Laplacian and Dirac operator, we design an energy-based projection test statistic. Empirical evaluation on real-world data—including foreign exchange arbitrage networks—demonstrates significant gains over graph-signal baselines and precise identification of anomalous transaction flows violating no-arbitrage conditions.

Technology Category

• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Interdisciplinary science discovery with web data mining
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Topological spaces, represented by simplicial complexes, 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. By leveraging the spectral dualities of Hodge and Dirac theory, practical topological 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 topological properties of the data. We demonstrate the effectiveness of the proposed topological detectors on various real-world data, including foreign currency exchange networks.

Problem

• Detect if simplicial complex signals lie in specific subspaces
• Identify anomalies disrupting subspace conditions in topological signals
• Propose energy-based detectors for single or multi-level simplicial data

Innovation

• Leverages Hodge and Dirac spectral dualities
• Proposes Neyman-Pearson matched subspace detectors
• Handles missing values with energy-based projection