Convolutional Filtering in Simplicial Complexes

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Abstract

This paper proposes convolutional filtering for data whose structure can be modeled by a simplicial complex (SC). SCs are mathematical tools that not only capture pairwise relationships as graphs but account also for higher-order network structures. These filters are built by following the shift-and-sum principle of the convolution operation and rely on the Hodge-Laplacians to shift the signal within the simplex. But since in SCs we have also inter-simplex coupling, we use the incidence matrices to transfer the signal in adjacent simplices and build a filter bank to jointly filter signals from different levels. We prove some interesting properties for the proposed filter bank, including permutation and orientation equivariance, a computational complexity that is linear in the SC dimension, and a spectral interpretation using the simplicial Fourier transform. We illustrate the proposed approach with numerical experiments.
Original languageEnglish
Title of host publicationProceedings of the ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
Place of PublicationPiscataway
PublisherIEEE
Pages5578-5582
Number of pages5
ISBN (Electronic)978-1-6654-0540-9
ISBN (Print)978-1-6654-0541-6
DOIs
Publication statusPublished - 2022
EventICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) - Singapore, Singapore
Duration: 23 May 202227 May 2022

Conference

ConferenceICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
Country/TerritorySingapore
CitySingapore
Period23/05/2227/05/22

Bibliographical note

Accepted author manuscript

Keywords

  • Hodge Laplacian
  • simplicial filter
  • topological signal processing

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