论文标题
简单复合物中的卷积过滤
Convolutional Filtering in Simplicial Complexes
论文作者
论文摘要
本文提出了可以通过简单复合物（SC）对结构进行建模的数据的卷积过滤。 SC是数学工具，不仅将成对关系捕获为图形，还可以说明高阶网络结构。这些过滤器是通过遵循卷积操作的移位原理来构建的，并依靠Hodge-Laplacians将信号转移到单纯形中。但是，由于在SC中，我们还具有简单的耦合，因此我们使用发射机矩阵将信号转移到相邻的简理中，并构建滤波器库以从不同级别共同过滤信号。我们证明了所提出的滤波器库的一些有趣属性，包括置换和方向均衡，在SC维度中线性线性的计算复杂性以及使用Simplicial Fourier变换的光谱解释。我们通过数值实验说明了所提出的方法。
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.
