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arXiv · 2201.12584

Convolutional Filtering in Simplicial Complexes

Elvin Isufi·Maosheng Yang
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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.

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Elvin Isufi, Maosheng Yang. 2022-01-29. Convolutional Filtering in Simplicial Complexes. https://arxiv.org/abs/2201.12584

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