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This research introduces an unoriented topological signal processing framework designed to handle higher-order signals without arbitrary orientation choices, addressing a key limitation in current flow-based methods. By deriving interaction-order decomposition and specialized regularizers, the study demonstrates improved reconstruction performance on real-world data where signal energy is unevenly distributed across orders. The work advances mathematical foundations for machine learning models operating on complex network structures, offering new tools for analyzing intricate data relationships.
Topological Signal Processing With Unoriented Operators
arXiv:2609.25310v1 Announce Type: new Abstract: Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.
Abstract:Topological signal processing (TSP) processes signals on simplicial complexes with oriented boundary operators, which is the natural choice for flow signals or when the topological invariants play a role for the task at hand. However, many higher-order signals carry no orientation, and applying oriented operators to them is not well-defined since it introduces an arbitrary choice of simplex orientation. We study an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices. First, we show that unoriented incidence and Laplacian matrices between arbitrary simplicial levels admit graph-like spectral properties. Second, since dropping orientation removes the Hodge decomposition, we introduce an unoriented counterpart, termed interaction-order decomposition, which quantifies how much of a higher-order signal is explained by aggregating lower-order signals. Third, we use this decomposition to derive regularizers for signal reconstruction that penalize each interaction order separately. Experiments on real-world data show that the order-aware regularizers outperform oriented baselines, with the largest gains when the signal energy is unevenly distributed across orders.
Machine Learning (cs.LG)
arXiv:2609.25310 [cs.LG]
(or arXiv:2609.25310v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.25310 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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