Convolutional Filtering in Simplicial Complexes

• URL: http://arxiv.org/abs/2201.12584v1
• Date: Sat, 29 Jan 2022 13:13:57 GMT
• Title: Convolutional Filtering in Simplicial Complexes
• Authors: Elvin Isufi and Maosheng Yang
• Abstract summary: 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.
• Score: 13.604803091781926
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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