# Matched Topological Subspace Detector

arXiv: https://arxiv.org/abs/2504.05892 · HTML: https://mlsanity.com/paper/2504.05892/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 33 references that are in the map.
- First posted on arXiv 8 Apr 2025.

## Builds on (most cited first, 16 of 16)

- [Topological Signal Processing over Simplicial Complexes](https://mlsanity.com/paper/1907.11577/) (2019)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond](https://mlsanity.com/paper/2101.05510/) (2021)
- [What are higher-order networks?](https://mlsanity.com/paper/2104.11329/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Flow Smoothing and Denoising: Graph Signal Processing in the Edge-Space](https://mlsanity.com/paper/1808.02111/) (2018)
- [Discrete Signal Processing on Graphs: Sampling Theory](https://mlsanity.com/paper/1503.05432/) (2015)
- [Generalized Simplicial Attention Neural Networks](https://mlsanity.com/paper/2309.02138/) (2023)
- [Graph-based Semi-Supervised & Active Learning for Edge Flows](https://mlsanity.com/paper/1905.07451/) (2019)
- [Simplicial Convolutional Filters](https://mlsanity.com/paper/2201.11720/) (2022)
- [Robust Graph Filter Identification and Graph Denoising from Signal Observations](https://mlsanity.com/paper/2210.08488/) (2022)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [The topological Dirac equation of networks and simplicial complexes](https://mlsanity.com/paper/2106.02929/) (2021)
- [Convolutional Filtering in Simplicial Complexes](https://mlsanity.com/paper/2201.12584/) (2022)
- [Weighted simplicial complexes and their representation power of higher-order network data and topology](https://mlsanity.com/paper/2207.04710/) (2022)
- [Higher-order signal processing with the Dirac operator](https://mlsanity.com/paper/2212.10196/) (2022)

## Built on by (most cited first, 3 of 3)

- [Stationarity and Spectral Characterization of Random Signals on Simplicial Complexes](https://mlsanity.com/paper/2602.03055/) (2026)
- [Topological Kalman Filtering on Cell Complexes](https://mlsanity.com/paper/2605.15955/) (2026)
- [Topological Signal Processing With Unoriented Operators](https://mlsanity.com/paper/2609.25310/) (2026)

## Abstract (arXiv)

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.