Signal Processing on Simplicial Complexes

Seby, Jean-Baptiste; Frantzen, Florian; Roddenberry, T. Mitchell; Zhu, Yu; Segarra, Santiago

doi:10.1007/978-3-030-91374-8_12

Cited by 15 publications

(9 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Most works in the literature so far (see, e.g., [34,35] for an overview) have considered this problem setup from a smoothing perspective, using the assumption that s is approximately harmonic. In this case, one can use an optimization formulation of the following form:…”

Section: A Problem Setup and Dirac Signal Processing Algorithmmentioning

confidence: 99%

“…Here a i are coefficients, which have to be chosen such that Q [n] is positive (semi-)definite to yield a well defined problem. Note that for a 2 = 1 and a j = 0 for j = 2 this reduces to the block-diagonal standard Hodge-Laplacian kernel L [12,35] (see Eq. ( 3)), and thus the topological signals of different dimension are filtered independently in this setting.…”

Section: A Problem Setup and Dirac Signal Processing Algorithmmentioning

confidence: 99%

“…In the last years signal processing on non-Euclidean domains such as graphs has received large attention [32]. Using variants of the Hodge Laplacian operators [19,33], these ideas have been extended to signal processing on simplicial complexes and other topological spaces [12,15,[34][35][36][37][38][39] and found their way into extensions of graphs neural networks [28,29,[40][41][42][43] Further current approaches build both on the Hodge Laplacians or the Magnetic Laplacian [44][45][46] to effectively extract information from higher-order or directed network data. However, despite great progress achieved so far, most current signal processing algorithms focus only on the filtering topological signals of a given dimension, e.g., flows on edges.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Dirac signal processing of higher-order topological signals

Calmon¹,

Bianconi²

2023

Preprint

View full text Add to dashboard Cite

We consider topological signals corresponding to variables supported on nodes, links and triangles of higher-order networks and simplicial complexes. So far such signals are typically processed independently of each other, and algorithms that can enforce a consistent processing of topological signals across different levels are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and (filled) triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is rooted in algebraic topology and formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order (Hodge) Laplacian matrix acting on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

show abstract

Section: A Problem Setup and Dirac Signal Processing Algorithmmentioning

confidence: 99%

Section: A Problem Setup and Dirac Signal Processing Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dirac signal processing of higher-order topological signals

Calmon¹,

Bianconi²

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Extending the tools of ordinary signal processing to graphs opened the door for a multitude of more combinatorially inspired applications [1,2,3,4]. Recently, this shift has been extended further to signal processing on topological spaces using simplicial complexes (SCs) [5,6,7,8,9], cellular complexes (CCs) [10,11,12], or cellular sheaves [13]. This move to topological signal processing allows both for a more general geometry of the underlying domain to be captured, as well as for additional constraints on the filtering to be taken into account.…”

Section: Introductionmentioning

confidence: 99%

Signal Processing on Product Spaces

Roddenberry¹,

Grande²,

Frantzen³

et al. 2023

Preprint

View full text Add to dashboard Cite

We establish a framework for signal processing on product spaces of simplicial and cellular complexes. For simplicity, we focus on the product of two complexes representing time and space, although our results generalize naturally to products of simplicial complexes of arbitrary dimension. Our framework leverages the structure of the eigenmodes of the Hodge Laplacian of the product space to jointly filter along time and space. To this end, we provide a decomposition theorem of the Hodge Laplacian of the product space, which highlights how the product structure induces a decomposition of each eigenmode into a spatial and temporal component. Finally, we apply our method to real world data, specifically for interpolating trajectories of buoys in the ocean from a limited set of observed trajectories.

show abstract

“…These models have been used in link prediction [26,44], optimal homology generator detection [52], and trajectory prediction [77]. This new research trend, collectively described as higher-order models, intertwines topological data analysis [32,24], topological signal processing [80,107,88,78,11,75,81] and geometric deep learning [109,23,36,63,16,70,15,54]. Current higher-order models have been defined on domains that allow modeling hierarchical and geometric-based relations (e.g., simplicial models [86,31,77,22,45] and cellular models [43]) or set-based relations (e.g., hypergraph models [98,106,50,4,35,37,73]).…”

Section: Introductionmentioning

confidence: 99%

Higher-Order Attention Networks

Hajij¹,

Papamarkou²,

Miolane³

et al. 2022

Preprint

View full text Add to dashboard Cite

This paper introduces higher-order attention networks (HOANs), a novel class of attention-based neural networks defined on a generalized higher-order domain called a combinatorial complex (CC). Similar to hypergraphs, CCs admit arbitrary set-like relations between a collection of abstract entities. Simultaneously, CCs permit the construction of hierarchical higher-order relations analogous to those supported by cell complexes. Thus, CCs effectively generalize both hypergraphs and cell complexes and combine their desirable characteristics. By exploiting the rich combinatorial nature of CCs, HOANs define a new class of messagepassing attention-based networks that unifies higher-order neural networks. Our evaluation of HOANs on tasks related to mesh shape analysis and graph learning demonstrates a competitive, and in some examples superior, predictive performance in comparison to state-of-the-art neural networks.

show abstract

Signal Processing on Simplicial Complexes

Cited by 15 publications

References 39 publications

Dirac signal processing of higher-order topological signals

Dirac signal processing of higher-order topological signals

Signal Processing on Product Spaces

Higher-Order Attention Networks

Contact Info

Product

Resources

About