Topological signal processing (TSP) traditionally uses oriented boundary operators to process signals on simplicial complexes, which is suitable for flow signals or when topological invariants are important. This paper introduces an unoriented TSP (UTSP) framework that replaces oriented boundaries with unoriented incidence matrices, demonstrating that these matrices retain graph-like spectral properties across simplicial levels. By removing orientation, the authors develop an interaction-order decomposition—an analogue to Hodge decomposition—to quantify how higher-order signals are explained by aggregating lower-order signals, and use this decomposition to create order-aware regularizers that outperform oriented baselines in real-world experiments, especially when signal energy is unevenly distributed across orders.
arXiv:2606. 17531v1 Announce Type: new Abstract: We investigate the learning of interpretable bases in non-negative matrix factorisation (NMF) by regularising the topology of the learned basis functions.
arXiv:2606. 09806v1 Announce Type: cross Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains.
The paper introduces higher-order positional encodings that enrich graph representations by incorporating topological information from lifted incidence structures, without altering existing graph learning backbones. It theoretically shows that these encodings can mix graph Laplacian frequencies beyond what scalar spectral filters achieve, and demonstrates their effectiveness on Graph Transformers for datasets like ZINC and synthetic benchmarks. The approach bridges graph positional encodings and topological deep learning, enabling standard models to exploit higher-order interactions.
arXiv:2606. 16990v1 Announce Type: new Abstract: While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales.
arXiv:2605. 13834v2 Announce Type: replace-cross Abstract: In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective.
arXiv:2601. 21207v4 Announce Type: replace-cross Abstract: Combinatorial and topological structures, such as graphs, simplicial complexes, and cell complexes, form the foundation of geometric and topological deep learning (GDL and TDL) architectures.
Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure. Processing such data requires a mathematical framework capable of simultaneously modeling heterogeneous local signal spaces and the transformations relating them.
arXiv:2607. 25295v2 Announce Type: replace Abstract: Tensorial multi-view clustering (TMC) has achieved strong performance due to its ability to capture high-order correlations across multiple views.
arXiv:2608. 15388v1 Announce Type: new Abstract: Topological deep learning (TDL) methods rely on lifting raw data into higher-order discrete domains such as simplicial complexes, cell complexes, and hypergraphs.
arXiv:2609.08152v1 Announce Type: new Abstract: Graph representation learning has largely focused on designing increasingly sophisticated models to transform graph topology into vector representation...
arXiv:2608. 01318v1 Announce Type: cross Abstract: Modern sensing, communication, and learning systems generate heterogeneous network signals, with local data differing in dimension, modality, and geometric structure.
arXiv:2609.05860v1 Announce Type: new Abstract: Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying...